Correspondence should be addressed to Stefan Tappe tappestochastikuni-hannoverde
Received 28 May 2013 Accepted 16 August 2013
Academic Editor Hong-Kun Xu
Copyright copy 2013 Stefan TappeThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
The goal of this review article is to provide a survey about the foundations of semilinear stochastic partial differential equations Inparticular we provide a detailed study of the concepts of strong weak and mild solutions establish their connections and reviewa standard existence and uniqueness result The proof of the existence result is based on a slightly extended version of the Banachfixed point theorem
1 Introduction
Semilinear stochastic partial differential equations (SPDEs)have a broad spectrum of applications including naturalsciences and economics The goal of this review article is toprovide a survey on the foundations of SPDEs which havebeen presented in the monographs [1ndash3] It may be beneficialfor students who are already aware about stochastic calculusin finite dimensions and who wish to have survey materialaccompanying the aforementioned references In particularwe review the relevant results from functional analysis aboutunbounded operators in Hilbert spaces and strongly contin-uous semigroups
A large part of this paper is devoted to a detailed study ofthe concepts of strong weak and mild solutions to SPDEsto establish their connections and to review and prove astandard existence and uniqueness result The proof of theexistence result is based on a slightly extended version of theBanach fixed point theorem
In the last part of this paper we study invariant manifoldsfor weak solutions to SPDEsThis topic does not belong to thegeneral theory of SPDEs but it uses and demonstrates manyof the results and techniques of the previous sections It arisesfrom the natural desire to express the solutions of SPDEswhich generally live in an infinite dimensional state spaceby means of a finite dimensional state process and thus toensure larger analytical tractability
This paper should also serve as an introductory studyto the general theory of SPDEs and it should enable thereader to learn about further topics and generalizations in thisfield Possible further directions are the study of martingalesolutions (see eg [1 3]) SPDEs with jumps (see eg [4] forSPDEs driven by the Levy processes and [5ndash8] for SPDEsdriven by Poisson random measures) and support theoremsas well as further invariance results for SPDEs see forexample [9 10]
The remainder of this paper is organized as follows InSections 2 and 3 we review the required results from func-tional analysis In particular we collect the relevant materialabout unbounded operators and strongly continuous semi-groups In Section 4 we review stochastic processes in infinitedimension In particular we recall the definition of a traceclass Wiener process and outline the construction of theIto integral In Section 5 we present the solution concepts forSPDEs and study their various connections In Section 6 wereview results about the regularity of stochastic convolutionintegrals which is essential for the study of mild solutionsto SPDEs In Section 7 we review a standard existence anduniqueness result Finally in Section 8 we deal with invariantmanifolds for weak solutions to SPDEs
2 Unbounded Operators in the Hilbert Spaces
In this section we review the relevant properties aboutunbounded operators We will start with operators in Banach
2 International Journal of Stochastic Analysis
spaces and focus on operators in Hilbert spaces later on Thereader can find the proofs of the upcoming results in anytextbook about functional analysis such as [11] or [12]
Let 119883 and 119884 be Banach spaces For a linear operator 119860
119883 sup D(119860) rarr 119884 defined on some subspace D(119860) of 119883 wecallD(119860) the domain of 119860
Definition 1 A linear operator 119860 119883 sup D(119860) rarr 119884 is calledclosed if for every sequence (119909
119899)119899isinN sub D(119860) such that the
limits 119909 = lim119899rarrinfin
119909119899
isin 119883 and 119910 = lim119899rarrinfin
119860119909119899
isin 119884 existone has 119909 isin D(119860) and 119860119909 = 119910
Definition 2 A linear operator 119860 119883 sup D(119860) rarr 119884 iscalled densely defined if its domainD(119860) is dense in 119883 thatisD(119860) = 119883
Definition 3 Let 119860 119883 sup D(119860) rarr 119883 be a linear operator
(1) The resolvent set of 119860 is defined as
120588 (119860) = 120582 isin C 120582 minus 119860 D (119860) 997888rarr 119883 is bijective and
(120582 minus 119860)minus1
isin 119871 (119883)
(1)
(2) The spectrum of 119860 is defined as 120590(119860) = C 120588(119860)(3) For 120582 isin 120588(119860) one defines the resolvent 119877(120582 119860) isin
119871(119883) as
119877 (120582 119860) = (120582 minus 119860)minus1
(2)
Now we will introduce the adjoint operator of a denselydefined operator in a Hilbert space Recall that for a boundedlinear operator 119879 isin 119871(119867
1 119867
2) mapping between two Hilbert
spaces1198671and119867
2 the adjoint operator is the unique bounded
linear operator 119879lowast
isin 119871(1198672 119867
1) such that
⟨119879119909 119910⟩1198672
= ⟨119909 119879lowast
119910⟩1198671
forall119909 isin 1198671 119910 isin 119867
2 (3)
In order to extend this definition to unbounded operatorsone recalls the following extension result for linear operators
Proposition 4 Let 119883 be a normed space let 119884 be a Banachspace let119863 sub 119883 be a dense subspace and letΦ 119863 rarr 119884 be acontinuous linear operatorThen there exists a unique continu-ous extension Φ 119883 rarr 119884 that is a continuous linear operatorwith Φ|
119863= Φ Moreover one has Φ = Φ
Now let119867 be aHilbert spaceWe recall the representationtheorem of Frechet-Riesz In the sequel the space119867
1015840 denotesthe dual space of 119867
Theorem 5 For every 1199091015840
isin 1198671015840 there exists a unique element
119909 isin 119867 with ⟨1199091015840
∙⟩ = ⟨119909 ∙⟩ In addition one has 119909 = 1199091015840
Let 119860 119867 sup D(119860) rarr 119867 be a densely defined operatorOne defines the subspace
) be arbitrary By virtue of the extension resultfor linear operators (Proposition 4) the operator
D (119860) 997891997888rarr R 119909 997891997888rarr ⟨119860119909 119910⟩ (5)
has a unique extension to a linear functional 1199111015840 isin 1198671015840 By the
representation theorem of Frechet-Riesz (Theorem 5) thereexists a unique element 119911 isin 119867 with ⟨119911
1015840
∙⟩ = ⟨119911 ∙⟩ Thisimplies that
⟨119860119909 119910⟩ = ⟨119909 119911⟩ forall119909 isin D (119860) (6)
Setting 119860lowast
119910 = 119911 this defines a linear operator 119860lowast
119867 sup
D(119860lowast
) rarr 119867 and one has
⟨119860119909 119910⟩ = ⟨119909 119860lowast
119910⟩ forall119909 isin D (119860) 119910 isin D (119860lowast
) (7)
Definition 6 The operator 119860lowast
119867 sup D(119860lowast
) rarr 119867 is calledthe adjoint operator of 119860
Proposition 7 Let 119860 119867 sup D(119860) rarr 119867 be densely definedand closed Then 119860
lowast is densely defined and one has 119860 = 119860lowastlowast
Lemma 8 Let 119867 be a separable Hilbert space and let 119860
119867 sup D(119860) rarr 119867 be a closed operator Then the domain(D(119860) sdot D(119860)
) endowed with the graph norm
119909D(119860)= (119909
2
+ 1198601199092
)12 (8)
is a separable Hilbert space too
3 Strongly Continuous Semigroups
In this section we present the required results about stronglycontinuous semigroups Concerning the proofs of theupcoming results the reader is referred to any textbook aboutfunctional analysis such as [11] or [12] Throughout this sec-tion let 119883 be a Banach space
Definition 9 Let (119878119905)119905ge0
be a family of continuous linearoperators 119878
119905 119883 rarr 119883 119905 ge 0
(1) The family (119878119905)119905ge0
is a called a strongly continuoussemigroup (or 119862
0-semigroup) if the following condi-
tions are satisfied
(i) 1198780= Id
(ii) 119878119904+119905
= 119878119904119878119905for all 119904 119905 ge 0
(iii) lim119905rarr0
119878119905119909 = 119909 for all 119909 isin 119883
(2) The family (119878119905)119905ge0
is called a norm continuous semi-group if the following conditions are satisfied
(i) 1198780= Id
(ii) 119878119904+119905
= 119878119904119878119905for all 119904 119905 ge 0
(iii) lim119905rarr0
119878119905minus Id = 0
Note that every norm continuous semigroup is also a 1198620-
semigroup The following growth estimate (9) will often beused when dealing with SPDEs
International Journal of Stochastic Analysis 3
Lemma 10 Let (119878119905)119905ge0
be a 1198620-semigroup Then there are
constants 119872 ge 1 and 120596 isin R such that1003817100381710038171003817119878119905
1003817100381710038171003817 le 119872119890120596119905
forall119905 ge 0 (9)
Definition 11 Let (119878119905)119905ge0
be a 1198620-semigroup
(1) The semigroup (119878119905)119905ge0
is called a semigroup of contrac-tions (or contractive) if
10038171003817100381710038171198781199051003817100381710038171003817 le 1 forall119905 ge 0 (10)
that is the growth estimate (9) is satisfied with119872 = 1
and 120596 = 0(2) The semigroup (119878
119905)119905ge0
is called a semigroup of pseu-docontractions (or pseudocontractive) if there exists aconstant 120596 isin R such that
10038171003817100381710038171198781199051003817100381710038171003817 le 119890
120596119905
forall119905 ge 0 (11)
that is the growth estimate (9) is satisfiedwith119872 = 1
If (119878119905)119905ge0
is a semigroup of pseudocontractions withgrowth estimate (11) then (119879
119905)119905ge0
given by
119879119905= 119890
minus120596119905
119878119905 119905 ge 0 (12)
is a semigroup of contractions Hence every pseudocon-tractive semigroup can be transformed into a semigroup ofcontractions which explains the term pseudocontractive
(2) For all 119909 isin 119883 and 119905 ge 0 one has int1199050
119878119904119909119889119904 isin D(119860) and
119860(int
119905
0
119878119904119909 119889119904) = 119878
119905119909 minus 119909 (26)
(3) For all 119909 isin D(119860) and 119905 ge 0 one has
int
119905
0
119878119904119860119909119889119904 = 119878
119905119909 minus 119909 (27)
The following result shows that the strongly continuoussemigroup (119878
119905)119905ge0
associated with generator119860 is uniqueThisexplains the term generator
Proposition 19 Two 1198620-semigroups (119878
119905)119905ge0
and (119879119905)119905ge0
withthe same infinitesimal generator 119860 coincide that is one has119878119905= 119879
119905for all 119905 ge 0
The next result characterizes all norm continuous semi-groups in terms of their generators
Proposition 20 Let (119878119905)119905ge0
be a 1198620-semigroup with infinitesi-
mal generator119860 Then the following statements are equivalent
(1) The semigroup (119878119905)119905ge0
is norm continuous(2) The operator 119860 is continuous(3) The domain of 119860 is given byD(119860) = 119883
If the previous conditions are satisfied then one has 119878119905= 119890
119905119860
for all 119905 ge 0
Now we are interested in characterizing all linear opera-tors 119860 which are the infinitesimal generator of some stronglycontinuous semigroup (119878
119905)119905ge0
The following theorem ofHille-Yosida gives a characterization in terms of the resolventwhich we have introduced in Definition 3
Theorem 21 (Hille-Yosida theorem) Let 119860 119883 sup D(119860) rarr
119883 be a linear operator and let119872 ge 1120596 isin R be constantsThenthe following statements are equivalent
(1) 119860 is the generator of a 1198620-semigroup (119878
119905)119905ge0
withgrowth estimate (9)
(2) 119860 is densely defined and closed and one has (120596infin) sub
1198991003817100381710038171003817 le 119872(120582 minus 120596)minus119899
forall120582 isin (120596infin) 119899 isin N (28)
In particular we obtain the following characterization ofthe generators of semigroups of contractions
Corollary 22 For a linear operator 119860 119883 sup D(119860) rarr 119883 thefollowing statements are equivalent
(1) 119860 is the generator of a semigroup (119878119905)119905ge0
of contractions(2) 119860 is densely defined and closed and one has (0infin) sub
120588(119860) and
119877 (120582 119860) le1
120582forall120582 isin (0infin) (29)
Proposition 23 Let (119878119905)119905ge0
be a 1198620-semigroup on 119883 with
generator 119860 Then the family (119878119905|D(119860)
)119905ge0
is a 1198620-semigroup
on (D(119860) sdot D(119860)) with generator 119860 D(119860
2
) sub D(119860) rarr
D(1198602
) where the domain is given by
D (1198602
) = 119909 isin D (119860) 119860119909 isin D (119860) (30)
Recall that we have introduced the adjoint operator foroperators in the Hilbert spaces in Definition 6
Proposition 24 Let 119867 be a Hilbert space and let (119878119905)119905ge0
be a 1198620-semigroup on 119867 with generator 119860 Then the family
of adjoint operators (119878lowast
119905)119905ge0
is a 1198620-semigroup on 119867 with
generator 119860lowast
4 Stochastic Processes in Infinite Dimension
In this section we recall the required foundations aboutstochastic processes in infinite dimension In particular werecall the definition of a trace class Wiener process and out-line the construction of the Ito integral
In the sequel (ΩF (F119905)119905ge0
P) denotes a filtered prob-ability space satisfying the usual conditions Let H be aseparable Hilbert space and let 119876 isin 119871(H) be a nuclear self-adjoint positive definite linear operator
Definition 25 A H-valued adapted continuous process 119882
is called a 119876-Wiener process if the following conditions aresatisfied
(i) One has 1198820= 0
(ii) The random variable 119882119905minus 119882
119904and the 120590-algebra F
119904
are independent for all 0 le 119904 le 119905(iii) One has 119882
119905minus 119882
119904sim 119873(0 (119905 minus 119904)119876) for all 0 le 119904 le 119905
In Definition 25 the distribution 119873(0 (119905 minus 119904)119876) is aGaussian measure with mean 0 and covariance operator (119905 minus
119904)119876 see for example [1 Section 232]The operator119876 is alsocalled the covariance operator of the Wiener process 119882 As119876 is a trace class operator we also call 119882 a trace class Wienerprocess
Now let 119882 be a 119876-Wiener process Then there exist anorthonormal basis (119890
119895)119895isinN of H and a sequence (120582
119895)119895isinN sub
(0infin) with sum119895isinN 120582
119895lt infin such that
119876119906 = sum
119895isinN
120582119895⟨119906 119890
119895⟩H119890119895 119906 isin H (31)
International Journal of Stochastic Analysis 5
Namely the 120582119895are the eigenvalues of 119876 and each 119890
119895is an
eigenvector corresponding to 120582119895 The space H
0= 119876
12
(H)equipped with the inner product
⟨119906 V⟩H0
= ⟨119876minus12
119906 119876minus12V⟩
H (32)
is another separable Hilbert space and (radic120582119895119890119895)119895isinN
is anorthonormal basis According to [1 Proposition 41] thesequence of stochastic processes (120573119895)
119895isinN defined as
120573119895
=1
radic120582119895
⟨119882 119890119895⟩H 119895 isin N (33)
is a sequence of real-valued independent standard Wienerprocesses and one has the expansion
119882 = sum
119895isinN
radic120582119895120573119895
119890119895 (34)
Now let us briefly sketch the construction of the Ito integralwith respect to the Wiener process 119882 Further details can befound in [1 3] We denote by 119871
0
2(119867) = 119871
2(H
0 119867) the space
of Hilbert-Schmidt operators from H0into119867 endowed with
the Hilbert-Schmidt norm
Φ1198710
2(119867)
= (sum
119895isinN
120582119895
10038171003817100381710038171003817Φ119890
119895
10038171003817100381710038171003817
2
)
12
Φ isin 1198710
2(119867) (35)
which itself is a separable Hilbert space The construction ofthe Ito integral is divided into three steps as follows
(1) For every 119871(H 119867)-valued simple process of the form
119883 = 119883010
+
119899
sum
119894=1
1198831198941(119905119894119905119894+1] (36)
with 0 = 1199051
lt sdot sdot sdot lt 119905119899+1
= 119879 and F119905119894
-measurablerandom variables 119883
119894 Ω rarr 119871(H 119867) for 119894 = 1 119899
we set
int
119905
0
119883119904119889119882
119904=
119899
sum
119894=1
119883119894(119882
119905and119905119894+1
minus 119882119905and119905119894
) (37)
(2) For every predictable 1198710
2(119867)-valued process 119883 satis-
fying
E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904] lt infin (38)
we extend the Ito integral int1199050
119883119904119889119882
119904by an extension
argument for linear operators In particular we obtainthe Ito isometry as follows
) is dense in119867 and hencewe obtain P-almost surely
119883119905and120591
= ℎ0+ int
119905and120591
0
(119860119883119904+ 120572 (119904 119883
119904)) 119889119904
+ int
119905and120591
0
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(53)
Consequently the process 119883 is also a local strong solution to(44) with lifetime 120591
Corollary 31 Let M sub D(119860) be a subset such that 119860 iscontinuous on M and let 119883 be a local weak solution to (44)with lifetime 120591 such that
119883119905and120591
isin M forall119905 ge 0 P-119886119897119898119900119904119905 119904119906119903119890119897119910 (54)
Then 119883 is also a local strong solution to (44) with lifetime 120591
Proof Since M sub D(119860) condition (54) implies that (46) isfulfilled Moreover by the continuity of 119860 onM the samplepaths of the process 119860119883 are P-almost surely continuous andhence we obtain (47) Consequently using Proposition 30the process 119883 is also a local strong solution to (44) withlifetime 120591
Proposition 32 Every strong solution119883 to (44) is also a mildsolution to (44)
Proof According to Lemma 8 the domain (D(119860) sdot D(119860))
endowed with the graph norm is a separable Hilbert spacetoo Hence by Lemma 18 for all 119905 ge 0 the function
119891 [0 119905] times D (119860) 997888rarr 119867 119891 (119904 119909) = 119878119905minus119904
119909 (55)
belongs to the class 11986212loc119887
([0 119905] times D(119860)119867) with partialderivatives
Since by Proposition 14 the domain D((119860lowast
)2
) is dense in(D(119860
lowast
) sdot D(119860lowast)) we get P-almost surely for all 120577 isin D(119860
lowast
)
the identity
⟨120577 119883119905⟩ = ⟨120577 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904⟩
(70)
Since by Proposition 14 the domainD(119860lowast
) is dense in119867 weobtain P-almost surely
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 (71)
proving that 119883 is a mild solution to (44)
Remark 36 Now the proof of Proposition 32 is an immediateconsequence of Propositions 29 and 35
We have just seen that every weak solution to (44) is also amild solution Under the following regularity condition (72)the converse of this statement holds true as well
Proposition 37 Let 119883 be a mild solution to (44) such that
120572 (119904 119883119904) minus 120572 (119904 119883
119904)⟩ 119889119904
= ⟨120577 int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904⟩ minus int
119905
0
⟨120577 120572 (119904 119883119904)⟩119889119904
(74)
Due to assumption (72) we may use Fubinirsquos theorem forstochastic integrals (see [3 Theorem 28]) which togetherwith Lemma 18 gives us P-almost surely
Due to assumption (80) we may use Fubinirsquos theorem forstochastic integrals (see [3 Theorem 28]) which togetherwith Lemma 18 gives us P-almost surely
int
119905
0
(119878119905minus119904
120590 (119904 119883119904) minus 120590 (119904 119883
This proves that 119883 is also a strong solution to (44)
The following result shows that for norm continuoussemigroups the concepts of strong weak and mild solutionsare equivalent In particular this applies for finite dimen-sional state spaces
Proposition 39 Suppose that the semigroup (119878119905)119905ge0
is normcontinuous Let 119883 be a stochastic process with 119883
0= ℎ
0 Then
the following statements are equivalent
(1) The process 119883 is a strong solution to (44)
(2) The process 119883 is a weak solution to (44)
(3) The process 119883 is a mild solution to (44)
Proof (1)rArr(2) This implication is a consequence ofProposition 29
(2)rArr(3) This implication is a consequence ofProposition 35
(3)rArr(1) By Proposition 20 we have 119860 isin 119871(119867) and 119878119905=
119890119905119860 119905 ge 0 Furthermore the family (119890
119905119860
)119905isinR is a 119862
0-group on
119867 Therefore and since 119883 is a mild solution to (44) we haveP-almost surely
(2) An element 119909 isin 119864 is called a fixed point of Φ if onehas
Φ (119909) = 119909 (109)
The following result is the well-known Banach fixed pointtheorem Its proof can be found for example in [13Theorem348]
Theorem 45 (The Banach fixed point theorem) Let 119864 be acomplete metric space and let Φ 119864 rarr 119864 be a contractionThen the mapping Φ has a unique fixed point
In this text we will use the following slight extension ofthe Banach fixed point theorem
Corollary 46 Let 119864 be a complete metric space and let Φ
119864 rarr 119864 be a mapping such that for some 119899 isin N the mappingΦ119899 is a contraction Then the mapping Φ has a unique fixed
point
Proof According to the Banach fixed point theorem(Theorem 45) the mapping Φ
119899 has a unique fixed point thatis there exists a unique element 119909 isin 119864 such that Φ119899
(119909) = 119909Therefore we have
Φ (119909) = Φ (Φ119899
(119909)) = Φ119899
(Φ (119909)) (110)
showing thatΦ(119909) is a fixedpoint ofΦ119899 SinceΦ119899 has a uniquefixed point we deduce thatΦ(119909) = 119909 showing that119909 is a fixedpoint of Φ
In order to prove uniqueness let 119910 isin 119864 be another fixedpoint ofΦ that is we haveΦ(119910) = 119910 By induction we obtain
showing that 119910 is a fixed point of Φ119899 Since the mapping Φ119899
has exactly one fixed point we obtain 119909 = 119910
An indispensable tool for proving uniqueness of mildsolutions to (44) will be the following version of Gronwallrsquosinequality see for example [14 Theorem 51]
Lemma 47 (Gronwallrsquos inequality) Let 119879 ge 0 be fixed let119891 [0 119879] rarr R
and hence by the Cauchy-Schwarz inequality the Ito isom-etry (39) the growth estimate (9) from Lemma 10 and thelocal Lipschitz conditions (113) we obtain
Using Gronwallrsquos inequality (see Lemma 47) we deduce that119891 equiv 0 Thus by the continuity of the sample paths of 119883 and119884 we obtain
P(⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ) = 1 forall119899 isin N (120)
and hence by the continuity of the probability measureP weconclude that
P(⋂
119905ge0
119883119905and120591
1Γ= 119884
119905and1205911Γ)
= P(⋂
119899isinN
⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ)
= lim119899rarrinfin
P(⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ) = 1
(121)
which completes the proof
The local Lipschitz conditions (113) are in general notsufficient in order to ensure the existence of mild solutions tothe SPDE (44) Now we will prove that the existence of mildsolutions follows from global Lipschitz and linear growthconditions on 120572 and 120590 For this we recall an auxiliary resultwhich extends the Ito isometry (39)
Lemma 49 Let 119879 ge 0 be arbitrary and let 119883 = (119883119905)119905isin[0119879]
for all 119905 ge 0 and all ℎ isin 119867 Then for every F0-measurable
random variable ℎ0
Ω rarr 119867 there exists a (up toindistinguishability) unique mild solution 119883 to (44)
Proof The uniqueness of mild solutions to (44) is a directconsequence of Theorem 48 and hence we may concentrateon the existence proof which we divide into the followingseveral steps
Step 1 First we suppose that the initial condition ℎ0satisfies
E[ℎ02119901
] lt infin for some 119901 gt 1 Let 119879 ge 0 be arbitrary Wedefine the Banach space
1198712119901
119879(119867) = 119871
2119901
(Ω times [0 119879] P119879P otimes 119889119905119867) (129)
and prove that the variation of constants equation
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(130)
has a unique solution in the space 1198712119901
119879(119867) This is done in the
following three steps
Step 11 For 119883 isin 1198712119901
119879(119867) we define the process Φ119883 by
(Φ119883)119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(131)
Then the process Φ119883 is well defined Indeed by the growthestimate (9) the linear growth condition (127) and Holderrsquosinequality we have
Consequently there exists an index 119899 isin N such that Φ119899 is
a contraction and hence according to the extension of theBanach fixed point theorem (see Corollary 46) the mappingΦ has a unique fixed point 119883 isin 119871
2119901
119879(119867) This fixed point 119883 is
a solution to the variation of constants equation (130) Since119879 ge 0 was arbitrary there exists a process 119883 which is a solu-tion of the variation of constants equation
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
119905 ge 0
(146)
Step 14 In order to prove that 119883 is a mild solution to (44)it remains to show that 119883 has a continuous version ByLemma 12 the process
119905 997891997888rarr 119878119905ℎ0 119905 ge 0 (147)
is continuous and by Proposition 41 the process
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 119905 ge 0 (148)
is continuous too Moreover for every 119879 ge 0 we have by thelinear growth condition (128) Holderrsquos inequality and since119883 isin 119871
Thus by Proposition 43 the stochastic convolution 120590 ⋆ 119882
given by
(120590 ⋆ 119882)119905= int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0 (150)
has a continuous version and consequently the process119883hasa continuous version too This continuous version is a mildsolution to (44)
Step 2 Now let ℎ0 Ω rarr 119867 be an arbitraryF
0-measurable
random variable We define the sequence (ℎ119899)119899isinN of F
0-
measurable random variables as
ℎ119899
0= ℎ
01ℎ0le119899
119899 isin N (151)
Let 119899 isin N be arbitrary Then as ℎ119899
0is bounded we have
E[ℎ119899
02119901
] lt infin for all 119901 gt 1 By Step 1 the SPDE
119889119883119899
119905= (119860119883
119899
119905+ 120572 (119905 119883
119899
119905)) 119889119905 + 120590 (119905 119883
119899
119905) 119889119882
119905
119883119899
0= ℎ
119899
0
(152)
18 International Journal of Stochastic Analysis
has amild solution119883119899We define the sequence (Ω
119899)119899isinN sub F
0
as
Ω119899=
1003817100381710038171003817ℎ01003817100381710038171003817 le 119899 119899 isin N (153)
Then we have Ω119899sub Ω
119898for 119899 le 119898 Ω = ⋃
119899isinN Ω119899 and
Ω119899sub ℎ
119899
0= ℎ
119898
0 sub ℎ
119899
0= ℎ
0 forall119899 le 119898 (154)
Thus byTheorem 48 we have (up to indistinguishability)
119883119899
1Ω119899
= 119883119898
1Ω119899
forall119899 le 119898 (155)
Consequently the process
119883 = lim119899rarrinfin
119883119899
1Ω119899
(156)
is a well-defined continuous and adapted process and wehave
119883119899
1Ω119899
= 119883119898
1Ω119899
= 1198831Ω119899
forall119899 le 119898 (157)
Furthermore we obtain P-almost surely
119883119905= lim
119899rarrinfin
119883119899
1199051Ω119899
= lim119899rarrinfin
1Ω119899
(119878119905ℎ119899
0+ int
119905
0
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+int
119905
0
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ119899
0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= lim119899rarrinfin
1Ω119899
(119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= 119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(158)
proving that 119883 is a mild solution to (44)
Remark 51 For the proof of Theorem 50 we have usedCorollary 46 which is a slight extension of the Banach fixedpoint theorem Such an idea has been applied for example in[15]
Remark 52 A recent method for proving existence anduniqueness of mild solutions to the SPDE (44) is the methodof the moving frame presented in [6] see also [8] It allowsto reduce SPDE problems to the study of SDEs in infinitedimension In order to apply this method we need that thesemigroup (119878
119905)119905ge0
is a semigroup of pseudocontractions
We close this section with a consequence about theexistence of weak solutions
Corollary 53 Suppose that conditions (125)ndash(128) are ful-filled Let ℎ
0 Ω rarr 119867 be aF
0-measurable random variable
such that E[ℎ02119901
] lt infin for some 119901 gt 1 Then there exists a(up to indistinguishability) unique weak solution 119883 to (44)
Proof According to Proposition 35 every weak solution119883 to(44) is also a mild solution to (44)Therefore the uniquenessof weak solutions to (44) is a consequence of Theorem 48
It remains to prove the existence of a weak solution to(44) Let 119879 ge 0 be arbitrary By Theorem 50 and its proofthere exists a mild solution119883 isin 119871
2119901
119879(119867) to (44) By the linear
growth condition (128) and Holderrsquos inequality we obtain
showing that condition (72) is fulfilled Thus byProposition 37 the process 119883 is also a weak solution to(44)
8 Invariant Manifolds for WeakSolutions to SPDEs
In this section we deal with invariant manifolds for time-homogeneous SPDEs of the type (44) This topic arises fromthe natural desire to express the solutions of the SPDE (44)which generally live in the infinite dimensional Hilbert space119867 by means of a finite dimensional state process and thusto ensure larger analytical tractability Our goal is to findconditions on the generator 119860 and the coefficients 120572 120590 suchthat for every starting point of a finite dimensional submani-fold the solution process stays on the submanifold
We start with the required preliminaries about finitedimensional submanifolds inHilbert spaces In the sequel let119867 be a separable Hilbert space
Definition 54 Let 119898 119896 isin N be positive integers A subsetM sub 119867 is called an 119898-dimensional 119862119896-submanifold of 119867
International Journal of Stochastic Analysis 19
if for every ℎ isin M there exist an open neighborhood 119880 sub 119867
of ℎ an open set 119881 sub R119898 and a mapping 120601 isin 1198622
(119881119867) suchthat
(1) the mapping 120601 119881 rarr 119880 cap M is a homeomorphism(2) for all 119910 isin 119881 the mapping 119863120601(119910) is injective
The mapping 120601 is called a parametrization ofM around ℎ
In what follows let M be an 119898-dimensional 119862119896-
submanifold of 119867
Lemma 55 Let 120601119894
119881119894
rarr 119880119894cap M 119894 = 1 2 be two
parametrizations with119882= 1198801cap119880
2capM = 0Then themapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (160)
is a 119862119896-diffeomorphism
Proof See [16 Lemma 611]
Corollary 56 Let ℎ isin M be arbitrary and let 120601119894
119881119894
rarr
119880119894capM 119894 = 1 2 be two parametrizations ofM around ℎ Then
one has
1198631206011(119910
1) (R
119898
) = 1198631206012(119910
2) (R
119898
) (161)
where 119910119894= 120601
minus1
119894(ℎ) for 119894 = 1 2
Proof Since1198801and119880
2are open neighborhoods of ℎ we have
119882= 1198801cap 119880
2cap M = 0 Thus by Lemma 55 the mapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (162)
is a 119862119896-diffeomorphism Using the chain rule we obtain
1198631206012(119910
2) (R
119898
)
= 119863 (1206011∘ (120601
minus1
1∘ 120601
2)) (119910
2) (R
119898
)
= 1198631206011(119910
1)119863 (120601
minus1
1∘ 120601
2) (119910
2) (R
119898
)
sub 1198631206011(119910
1) (R
119898
)
(163)
and analogously we prove that 1198631206011(119910
1)(R119898
) sub
1198631206012(119910
2)(R119898
)
Definition 57 Let ℎ isin M be arbitraryThe tangent space ofMto ℎ is the subspace
119879ℎM = 119863120601 (119910) (R
119898
) (164)
where 119910 = 120601minus1
(ℎ) and 120601 119881 rarr 119880 cap M denotes aparametrization ofM around ℎ
Remark 58 Note that according to Corollary 56 theDefinition 57 of the tangent space 119879
ℎM does not depend on
the choice of the parametrization 120601 119881 rarr 119880 cap M
Proposition 59 Let ℎ isin M be arbitrary and let 120601 119881 rarr
119880 cap M be a parametrization ofM around ℎ Then there existan open set 119881
0sub 119881 an open neighborhood 119880
0sub 119880 of ℎ and
a mapping 120601 isin 119862119896
119887(R119898
119867) with 120601|1198810
= 120601|1198810
such that 120601|1198810
1198810
rarr 1198800cap M is a parametrization ofM around ℎ too
Proof See [16 Remark 611]
Remark 60 By Proposition 59 we may assume that anyparametrization 120601 119881 rarr 119880 cap M has an extension 120601 isin
119862119896
119887(R119898
119867)
Proposition 61 Let 119863 sub 119867 be a dense subset For every ℎ0isin
M there exist 1205771 120577
119898isin 119863 and a parametrization 120601 119881 rarr
119880 cap M around ℎ0such that
120601 (⟨120577 ℎ⟩) = ℎ forallℎ isin 119880 cap M (165)
where one uses the notation ⟨120577 ℎ⟩ = (⟨1205771 ℎ⟩ ⟨120577
119898 ℎ⟩) isin
R119898
Proof See [16 Proposition 612]
Proposition 62 Let 120601 119881 rarr 119880 cap M be a parametrizationas in Proposition 61 Then the following statements are true
(1) The elements 1205771 120577
119898are linearly independent in 119867
(2) For every ℎ isin 119880 cap M one has the direct sum decom-position
Therefore by Proposition 64 for every 1199100isin R119898 there exists
a (up to indistinguishability) unique weak solution to (192)which according to Proposition 39 is also a strong solutionto (192) The uniqueness of strong solutions to (192) is a con-sequence of Proposition 39 andTheorem 48
Now we are ready to formulate and prove ourmain resultof this section
Theorem 68 The following statements are equivalent
(1) The submanifoldM is locally invariant for (176)(2) One has
M sub D (119860) (195)
120590119895
(ℎ) isin 119879ℎM forallℎ isin M 119886119897119897 119895 isin N (196)
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) isin 119879ℎM forallℎ isin M (197)
(3) The operator 119860 is continuous onM and for each ℎ0isin
M there exists a local strong solution 119883 to (176) withsome lifetime 120591 gt 0 such that
119883119905and120591
isin M forall119905 ge 0 P-119886119897119898119900119904119905 119904119906119903119890119897119910 (198)
Proof (1)rArr(2) Let ℎ isin M be arbitrary By Proposition 61and Remark 60 there exist elements 120577
1 120577
119898isin D(119860
lowast
) anda parametrization 120601 119881 rarr 119880 cap M around ℎ such that theinverse 120601
minus1
119880 cap M rarr 119881 is given by 120601minus1
= ⟨120577 ∙⟩ and 120601
has an extension 120601 isin 1198622
119887(R119898
119867) Since the submanifoldM islocally invariant for (176) there exists a local weak solution119883
to (176) with initial condition ℎ and some lifetime 984858 gt 0 suchthat
119883119905and984858
isin M forall119905 ge 0 P-almost surely (199)
Since119880 is an open neighborhood of ℎ there exists 120598 gt 0 suchthat 119861
120598(ℎ) sub 119880 where 119861
120598(ℎ) denotes the open ball
119861120598(ℎ) = 119892 isin 119867
1003817100381710038171003817119892 minus ℎ1003817100381710038171003817 lt 120598 (200)
We define the stopping time
120591 = 984858 and inf 119905 ge 0 119883119905notin 119861
120598(ℎ) (201)
Since the process119883 has continuous sample paths and satisfies1198830= ℎ we have 120591 gt 0 and P-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (202)
22 International Journal of Stochastic Analysis
Defining the R119898-valued process 119884 = ⟨120577 119883⟩ we have P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (203)
Moreover since 119883 is a weak solution to (176) with initialcondition ℎ setting 119910 = ⟨120577 ℎ⟩ isin 119881 we have P-almost surely
119884119905and120591
= ⟨120577 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120577 120590119895
(119883119904)⟩ 119889120573
119895
119904
= ⟨120577 ℎ⟩ + int
119905and120591
0
120572120601120577
(⟨120577 119883119904⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(⟨120577 119883
119904⟩) 119889120573
119895
119904
= 119910 + int
119905and120591
0
120572120601120577
(119884119904) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(204)
showing that 119884 is a local strong solution to (192) with initialcondition 119910 By Itorsquos formula (Theorem 26) we obtain P-almost surely
119883119905and120591
= 120601 (119884119905and120591
)
= ℎ + int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904)))119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(205)
Now let 120585 isin D(119860lowast
) be arbitrary Then we have P-almostsurely
⟨120585 119883119905and120591
⟩
= ⟨120585 ℎ⟩
+ int
119905and120591
0
⟨120585119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 119863120601 (119884119904) 120590
119895
120601120577(119884
119904)⟩119889120573
119895
119904 119905 ge 0
(206)
On the other hand since 119883 is a local weak solution to (176)with initial condition ℎ and lifetime 120591 we have P-almostsurely for all 119905 ge 0 the identity
⟨120585 119883119905and120591
⟩ = ⟨120585 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904)⟩ 119889120573
119895
119904
(207)
Combining (206) and (207) yields up to indistinguishability
119861 + 119872 = 0 (208)
where the processes 119861 and 119872 are defined as
119861119905= int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩) 119889119904
119905 ge 0
119872119905= sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩ 119889120573
119895
119904
119905 ge 0
(209)
The process 119861 + 119872 is a continuous semimartingale withcanonical decomposition (208) Since the canonical decom-position of a continuous semimartingale is unique up toindistinguishability we deduce that 119861 = 119872 = 0 up toindistinguishability Using the Ito isometry (39) we obtain P-almost surely
int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩
+ ⟨120585 120572 (119883119904) minus 119863120601 (119884
showing that119883 is a local strong solution to (44) with lifetime120591
(3)rArr(1) This implication is a direct consequence ofProposition 29
The results from this section are closely related to the exis-tence of finite dimensional realizations that is the existenceof invariantmanifolds for each starting point ℎ
0 andwe point
out the papers [17ndash22] regarding this topic Furthermore wemention thatTheorem 68 has been extended in [23] to SPDEswith jumps
Acknowledgments
The author is grateful to Daniel Gaigall Georg GrafendorferFlorian Modler and Thomas Salfeld for the valuable com-ments and discussions
References
[1] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions CambridgeUniversity Press CambridgeUK 1992
[2] C Prevot and M Rockner A Concise Course on StochasticPartial Differential Equations Springer Berlin Germany 2007
[3] L Gawarecki and V Mandrekar Stochastic Differential Equa-tions in Infinite Dimensions with Applications to StochasticPartial Differential Equations Springer Berlin Germany 2011
[4] S Peszat and J Zabczyk Stochastic Partial Differential Equationswith Levy Noise Cambridge University Press Cambridge UK2007
[5] S Albeverio V Mandrekar and B Rudiger ldquoExistence of mildsolutions for stochastic differential equations and semilinearequations with non-Gaussian Levy noiserdquo Stochastic Processesand Their Applications vol 119 no 3 pp 835ndash863 2009
[6] D Filipovic S Tappe and J Teichmann ldquoJump-diffusions inHilbert spaces existence stability and numericsrdquo Stochasticsvol 82 no 5 pp 475ndash520 2010
[7] C Marinelli C Prevot and M Rockner ldquoRegular dependenceon initial data for stochastic evolution equations with multi-plicative Poisson noiserdquo Journal of Functional Analysis vol 258no 2 pp 616ndash649 2010
[8] S Tappe ldquoSome refinements of existence results for SPDEsdriven by Wiener processes and Poisson random measuresrdquoInternational Journal of Stochastic Analysis vol 2012 Article ID236327 24 pages 2012
[9] T Nakayama ldquoSupport theorem for mild solutions of SDErsquos inHilbert spacesrdquo Journal of Mathematical Sciences the Universityof Tokyo vol 11 no 3 pp 245ndash311 2004
[10] T Nakayama ldquoViability theorem for SPDErsquos including HJMframeworkrdquo Journal of Mathematical Sciences the University ofTokyo vol 11 no 3 pp 313ndash324 2004
[11] W Rudin Functional Analysis McGraw-Hill New York NYUSA 2nd edition 1991
[12] DWerner Funktionalanalysis Springer Berlin Germany 2007[13] C D Aliprantis and K C Border Infinite Dimensional Analysis
Springer Berlin Germany 3rd edition 2006[14] S N Ethier and T G KurtzMarkov Processes Characterization
andConvergenceWiley Series in Probability and Statistics JohnWiley amp Sons New Jersey NJ USA 1986
[15] V Mandrekar and B Rudiger ldquoExistence and uniqueness ofpath wise solutions for stochastic integral equations driven byLevy noise on separable Banach spacesrdquo Stochastics vol 78 no4 pp 189ndash212 2006
[16] D Filipovic Consistency Problems for Heath-Jarrow-MortonInterest Rate Models Springer Berlin Germany 2001
[17] T Bjork and L Svensson ldquoOn the existence of finite-dimensional realizations for nonlinear forward rate modelsrdquoMathematical Finance vol 11 no 2 pp 205ndash243 2001
[18] T Bjork and C Landen ldquoOn the construction of finite dimen-sional realizations for nonlinear forward rate modelsrdquo Financeand Stochastics vol 6 no 3 pp 303ndash331 2002
[19] D Filipovic and J Teichmann ldquoExistence of invariantmanifoldsfor stochastic equations in infinite dimensionrdquo Journal ofFunctional Analysis vol 197 no 2 pp 398ndash432 2003
[20] D Filipovic and J Teichmann ldquoOn the geometry of the termstructure of interest ratesrdquo Proceedings of The Royal Society ofLondon A vol 460 no 2041 pp 129ndash167 2004
[21] S Tappe ldquoAn alternative approach on the existence of affinerealizations for HJM term structure modelsrdquo Proceedings of
International Journal of Stochastic Analysis 25
TheRoyal Society of LondonA vol 466 no 2122 pp 3033ndash30602010
[22] S Tappe ldquoExistence of affine realizations for Levy term struc-ture modelsrdquo Proceedings of The Royal Society of London A vol468 no 2147 pp 3685ndash3704 2012
[23] D Filipovic S Tappe and J Teichmann ldquoInvariant manifoldswith boundary for jump-diffusionsrdquo httparxivorgabs12021076
spaces and focus on operators in Hilbert spaces later on Thereader can find the proofs of the upcoming results in anytextbook about functional analysis such as [11] or [12]
Let 119883 and 119884 be Banach spaces For a linear operator 119860
119883 sup D(119860) rarr 119884 defined on some subspace D(119860) of 119883 wecallD(119860) the domain of 119860
Definition 1 A linear operator 119860 119883 sup D(119860) rarr 119884 is calledclosed if for every sequence (119909
119899)119899isinN sub D(119860) such that the
limits 119909 = lim119899rarrinfin
119909119899
isin 119883 and 119910 = lim119899rarrinfin
119860119909119899
isin 119884 existone has 119909 isin D(119860) and 119860119909 = 119910
Definition 2 A linear operator 119860 119883 sup D(119860) rarr 119884 iscalled densely defined if its domainD(119860) is dense in 119883 thatisD(119860) = 119883
Definition 3 Let 119860 119883 sup D(119860) rarr 119883 be a linear operator
(1) The resolvent set of 119860 is defined as
120588 (119860) = 120582 isin C 120582 minus 119860 D (119860) 997888rarr 119883 is bijective and
(120582 minus 119860)minus1
isin 119871 (119883)
(1)
(2) The spectrum of 119860 is defined as 120590(119860) = C 120588(119860)(3) For 120582 isin 120588(119860) one defines the resolvent 119877(120582 119860) isin
119871(119883) as
119877 (120582 119860) = (120582 minus 119860)minus1
(2)
Now we will introduce the adjoint operator of a denselydefined operator in a Hilbert space Recall that for a boundedlinear operator 119879 isin 119871(119867
1 119867
2) mapping between two Hilbert
spaces1198671and119867
2 the adjoint operator is the unique bounded
linear operator 119879lowast
isin 119871(1198672 119867
1) such that
⟨119879119909 119910⟩1198672
= ⟨119909 119879lowast
119910⟩1198671
forall119909 isin 1198671 119910 isin 119867
2 (3)
In order to extend this definition to unbounded operatorsone recalls the following extension result for linear operators
Proposition 4 Let 119883 be a normed space let 119884 be a Banachspace let119863 sub 119883 be a dense subspace and letΦ 119863 rarr 119884 be acontinuous linear operatorThen there exists a unique continu-ous extension Φ 119883 rarr 119884 that is a continuous linear operatorwith Φ|
119863= Φ Moreover one has Φ = Φ
Now let119867 be aHilbert spaceWe recall the representationtheorem of Frechet-Riesz In the sequel the space119867
1015840 denotesthe dual space of 119867
Theorem 5 For every 1199091015840
isin 1198671015840 there exists a unique element
119909 isin 119867 with ⟨1199091015840
∙⟩ = ⟨119909 ∙⟩ In addition one has 119909 = 1199091015840
Let 119860 119867 sup D(119860) rarr 119867 be a densely defined operatorOne defines the subspace
) be arbitrary By virtue of the extension resultfor linear operators (Proposition 4) the operator
D (119860) 997891997888rarr R 119909 997891997888rarr ⟨119860119909 119910⟩ (5)
has a unique extension to a linear functional 1199111015840 isin 1198671015840 By the
representation theorem of Frechet-Riesz (Theorem 5) thereexists a unique element 119911 isin 119867 with ⟨119911
1015840
∙⟩ = ⟨119911 ∙⟩ Thisimplies that
⟨119860119909 119910⟩ = ⟨119909 119911⟩ forall119909 isin D (119860) (6)
Setting 119860lowast
119910 = 119911 this defines a linear operator 119860lowast
119867 sup
D(119860lowast
) rarr 119867 and one has
⟨119860119909 119910⟩ = ⟨119909 119860lowast
119910⟩ forall119909 isin D (119860) 119910 isin D (119860lowast
) (7)
Definition 6 The operator 119860lowast
119867 sup D(119860lowast
) rarr 119867 is calledthe adjoint operator of 119860
Proposition 7 Let 119860 119867 sup D(119860) rarr 119867 be densely definedand closed Then 119860
lowast is densely defined and one has 119860 = 119860lowastlowast
Lemma 8 Let 119867 be a separable Hilbert space and let 119860
119867 sup D(119860) rarr 119867 be a closed operator Then the domain(D(119860) sdot D(119860)
) endowed with the graph norm
119909D(119860)= (119909
2
+ 1198601199092
)12 (8)
is a separable Hilbert space too
3 Strongly Continuous Semigroups
In this section we present the required results about stronglycontinuous semigroups Concerning the proofs of theupcoming results the reader is referred to any textbook aboutfunctional analysis such as [11] or [12] Throughout this sec-tion let 119883 be a Banach space
Definition 9 Let (119878119905)119905ge0
be a family of continuous linearoperators 119878
119905 119883 rarr 119883 119905 ge 0
(1) The family (119878119905)119905ge0
is a called a strongly continuoussemigroup (or 119862
0-semigroup) if the following condi-
tions are satisfied
(i) 1198780= Id
(ii) 119878119904+119905
= 119878119904119878119905for all 119904 119905 ge 0
(iii) lim119905rarr0
119878119905119909 = 119909 for all 119909 isin 119883
(2) The family (119878119905)119905ge0
is called a norm continuous semi-group if the following conditions are satisfied
(i) 1198780= Id
(ii) 119878119904+119905
= 119878119904119878119905for all 119904 119905 ge 0
(iii) lim119905rarr0
119878119905minus Id = 0
Note that every norm continuous semigroup is also a 1198620-
semigroup The following growth estimate (9) will often beused when dealing with SPDEs
International Journal of Stochastic Analysis 3
Lemma 10 Let (119878119905)119905ge0
be a 1198620-semigroup Then there are
constants 119872 ge 1 and 120596 isin R such that1003817100381710038171003817119878119905
1003817100381710038171003817 le 119872119890120596119905
forall119905 ge 0 (9)
Definition 11 Let (119878119905)119905ge0
be a 1198620-semigroup
(1) The semigroup (119878119905)119905ge0
is called a semigroup of contrac-tions (or contractive) if
10038171003817100381710038171198781199051003817100381710038171003817 le 1 forall119905 ge 0 (10)
that is the growth estimate (9) is satisfied with119872 = 1
and 120596 = 0(2) The semigroup (119878
119905)119905ge0
is called a semigroup of pseu-docontractions (or pseudocontractive) if there exists aconstant 120596 isin R such that
10038171003817100381710038171198781199051003817100381710038171003817 le 119890
120596119905
forall119905 ge 0 (11)
that is the growth estimate (9) is satisfiedwith119872 = 1
If (119878119905)119905ge0
is a semigroup of pseudocontractions withgrowth estimate (11) then (119879
119905)119905ge0
given by
119879119905= 119890
minus120596119905
119878119905 119905 ge 0 (12)
is a semigroup of contractions Hence every pseudocon-tractive semigroup can be transformed into a semigroup ofcontractions which explains the term pseudocontractive
(2) For all 119909 isin 119883 and 119905 ge 0 one has int1199050
119878119904119909119889119904 isin D(119860) and
119860(int
119905
0
119878119904119909 119889119904) = 119878
119905119909 minus 119909 (26)
(3) For all 119909 isin D(119860) and 119905 ge 0 one has
int
119905
0
119878119904119860119909119889119904 = 119878
119905119909 minus 119909 (27)
The following result shows that the strongly continuoussemigroup (119878
119905)119905ge0
associated with generator119860 is uniqueThisexplains the term generator
Proposition 19 Two 1198620-semigroups (119878
119905)119905ge0
and (119879119905)119905ge0
withthe same infinitesimal generator 119860 coincide that is one has119878119905= 119879
119905for all 119905 ge 0
The next result characterizes all norm continuous semi-groups in terms of their generators
Proposition 20 Let (119878119905)119905ge0
be a 1198620-semigroup with infinitesi-
mal generator119860 Then the following statements are equivalent
(1) The semigroup (119878119905)119905ge0
is norm continuous(2) The operator 119860 is continuous(3) The domain of 119860 is given byD(119860) = 119883
If the previous conditions are satisfied then one has 119878119905= 119890
119905119860
for all 119905 ge 0
Now we are interested in characterizing all linear opera-tors 119860 which are the infinitesimal generator of some stronglycontinuous semigroup (119878
119905)119905ge0
The following theorem ofHille-Yosida gives a characterization in terms of the resolventwhich we have introduced in Definition 3
Theorem 21 (Hille-Yosida theorem) Let 119860 119883 sup D(119860) rarr
119883 be a linear operator and let119872 ge 1120596 isin R be constantsThenthe following statements are equivalent
(1) 119860 is the generator of a 1198620-semigroup (119878
119905)119905ge0
withgrowth estimate (9)
(2) 119860 is densely defined and closed and one has (120596infin) sub
1198991003817100381710038171003817 le 119872(120582 minus 120596)minus119899
forall120582 isin (120596infin) 119899 isin N (28)
In particular we obtain the following characterization ofthe generators of semigroups of contractions
Corollary 22 For a linear operator 119860 119883 sup D(119860) rarr 119883 thefollowing statements are equivalent
(1) 119860 is the generator of a semigroup (119878119905)119905ge0
of contractions(2) 119860 is densely defined and closed and one has (0infin) sub
120588(119860) and
119877 (120582 119860) le1
120582forall120582 isin (0infin) (29)
Proposition 23 Let (119878119905)119905ge0
be a 1198620-semigroup on 119883 with
generator 119860 Then the family (119878119905|D(119860)
)119905ge0
is a 1198620-semigroup
on (D(119860) sdot D(119860)) with generator 119860 D(119860
2
) sub D(119860) rarr
D(1198602
) where the domain is given by
D (1198602
) = 119909 isin D (119860) 119860119909 isin D (119860) (30)
Recall that we have introduced the adjoint operator foroperators in the Hilbert spaces in Definition 6
Proposition 24 Let 119867 be a Hilbert space and let (119878119905)119905ge0
be a 1198620-semigroup on 119867 with generator 119860 Then the family
of adjoint operators (119878lowast
119905)119905ge0
is a 1198620-semigroup on 119867 with
generator 119860lowast
4 Stochastic Processes in Infinite Dimension
In this section we recall the required foundations aboutstochastic processes in infinite dimension In particular werecall the definition of a trace class Wiener process and out-line the construction of the Ito integral
In the sequel (ΩF (F119905)119905ge0
P) denotes a filtered prob-ability space satisfying the usual conditions Let H be aseparable Hilbert space and let 119876 isin 119871(H) be a nuclear self-adjoint positive definite linear operator
Definition 25 A H-valued adapted continuous process 119882
is called a 119876-Wiener process if the following conditions aresatisfied
(i) One has 1198820= 0
(ii) The random variable 119882119905minus 119882
119904and the 120590-algebra F
119904
are independent for all 0 le 119904 le 119905(iii) One has 119882
119905minus 119882
119904sim 119873(0 (119905 minus 119904)119876) for all 0 le 119904 le 119905
In Definition 25 the distribution 119873(0 (119905 minus 119904)119876) is aGaussian measure with mean 0 and covariance operator (119905 minus
119904)119876 see for example [1 Section 232]The operator119876 is alsocalled the covariance operator of the Wiener process 119882 As119876 is a trace class operator we also call 119882 a trace class Wienerprocess
Now let 119882 be a 119876-Wiener process Then there exist anorthonormal basis (119890
119895)119895isinN of H and a sequence (120582
119895)119895isinN sub
(0infin) with sum119895isinN 120582
119895lt infin such that
119876119906 = sum
119895isinN
120582119895⟨119906 119890
119895⟩H119890119895 119906 isin H (31)
International Journal of Stochastic Analysis 5
Namely the 120582119895are the eigenvalues of 119876 and each 119890
119895is an
eigenvector corresponding to 120582119895 The space H
0= 119876
12
(H)equipped with the inner product
⟨119906 V⟩H0
= ⟨119876minus12
119906 119876minus12V⟩
H (32)
is another separable Hilbert space and (radic120582119895119890119895)119895isinN
is anorthonormal basis According to [1 Proposition 41] thesequence of stochastic processes (120573119895)
119895isinN defined as
120573119895
=1
radic120582119895
⟨119882 119890119895⟩H 119895 isin N (33)
is a sequence of real-valued independent standard Wienerprocesses and one has the expansion
119882 = sum
119895isinN
radic120582119895120573119895
119890119895 (34)
Now let us briefly sketch the construction of the Ito integralwith respect to the Wiener process 119882 Further details can befound in [1 3] We denote by 119871
0
2(119867) = 119871
2(H
0 119867) the space
of Hilbert-Schmidt operators from H0into119867 endowed with
the Hilbert-Schmidt norm
Φ1198710
2(119867)
= (sum
119895isinN
120582119895
10038171003817100381710038171003817Φ119890
119895
10038171003817100381710038171003817
2
)
12
Φ isin 1198710
2(119867) (35)
which itself is a separable Hilbert space The construction ofthe Ito integral is divided into three steps as follows
(1) For every 119871(H 119867)-valued simple process of the form
119883 = 119883010
+
119899
sum
119894=1
1198831198941(119905119894119905119894+1] (36)
with 0 = 1199051
lt sdot sdot sdot lt 119905119899+1
= 119879 and F119905119894
-measurablerandom variables 119883
119894 Ω rarr 119871(H 119867) for 119894 = 1 119899
we set
int
119905
0
119883119904119889119882
119904=
119899
sum
119894=1
119883119894(119882
119905and119905119894+1
minus 119882119905and119905119894
) (37)
(2) For every predictable 1198710
2(119867)-valued process 119883 satis-
fying
E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904] lt infin (38)
we extend the Ito integral int1199050
119883119904119889119882
119904by an extension
argument for linear operators In particular we obtainthe Ito isometry as follows
) is dense in119867 and hencewe obtain P-almost surely
119883119905and120591
= ℎ0+ int
119905and120591
0
(119860119883119904+ 120572 (119904 119883
119904)) 119889119904
+ int
119905and120591
0
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(53)
Consequently the process 119883 is also a local strong solution to(44) with lifetime 120591
Corollary 31 Let M sub D(119860) be a subset such that 119860 iscontinuous on M and let 119883 be a local weak solution to (44)with lifetime 120591 such that
119883119905and120591
isin M forall119905 ge 0 P-119886119897119898119900119904119905 119904119906119903119890119897119910 (54)
Then 119883 is also a local strong solution to (44) with lifetime 120591
Proof Since M sub D(119860) condition (54) implies that (46) isfulfilled Moreover by the continuity of 119860 onM the samplepaths of the process 119860119883 are P-almost surely continuous andhence we obtain (47) Consequently using Proposition 30the process 119883 is also a local strong solution to (44) withlifetime 120591
Proposition 32 Every strong solution119883 to (44) is also a mildsolution to (44)
Proof According to Lemma 8 the domain (D(119860) sdot D(119860))
endowed with the graph norm is a separable Hilbert spacetoo Hence by Lemma 18 for all 119905 ge 0 the function
119891 [0 119905] times D (119860) 997888rarr 119867 119891 (119904 119909) = 119878119905minus119904
119909 (55)
belongs to the class 11986212loc119887
([0 119905] times D(119860)119867) with partialderivatives
Since by Proposition 14 the domain D((119860lowast
)2
) is dense in(D(119860
lowast
) sdot D(119860lowast)) we get P-almost surely for all 120577 isin D(119860
lowast
)
the identity
⟨120577 119883119905⟩ = ⟨120577 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904⟩
(70)
Since by Proposition 14 the domainD(119860lowast
) is dense in119867 weobtain P-almost surely
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 (71)
proving that 119883 is a mild solution to (44)
Remark 36 Now the proof of Proposition 32 is an immediateconsequence of Propositions 29 and 35
We have just seen that every weak solution to (44) is also amild solution Under the following regularity condition (72)the converse of this statement holds true as well
Proposition 37 Let 119883 be a mild solution to (44) such that
120572 (119904 119883119904) minus 120572 (119904 119883
119904)⟩ 119889119904
= ⟨120577 int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904⟩ minus int
119905
0
⟨120577 120572 (119904 119883119904)⟩119889119904
(74)
Due to assumption (72) we may use Fubinirsquos theorem forstochastic integrals (see [3 Theorem 28]) which togetherwith Lemma 18 gives us P-almost surely
Due to assumption (80) we may use Fubinirsquos theorem forstochastic integrals (see [3 Theorem 28]) which togetherwith Lemma 18 gives us P-almost surely
int
119905
0
(119878119905minus119904
120590 (119904 119883119904) minus 120590 (119904 119883
This proves that 119883 is also a strong solution to (44)
The following result shows that for norm continuoussemigroups the concepts of strong weak and mild solutionsare equivalent In particular this applies for finite dimen-sional state spaces
Proposition 39 Suppose that the semigroup (119878119905)119905ge0
is normcontinuous Let 119883 be a stochastic process with 119883
0= ℎ
0 Then
the following statements are equivalent
(1) The process 119883 is a strong solution to (44)
(2) The process 119883 is a weak solution to (44)
(3) The process 119883 is a mild solution to (44)
Proof (1)rArr(2) This implication is a consequence ofProposition 29
(2)rArr(3) This implication is a consequence ofProposition 35
(3)rArr(1) By Proposition 20 we have 119860 isin 119871(119867) and 119878119905=
119890119905119860 119905 ge 0 Furthermore the family (119890
119905119860
)119905isinR is a 119862
0-group on
119867 Therefore and since 119883 is a mild solution to (44) we haveP-almost surely
(2) An element 119909 isin 119864 is called a fixed point of Φ if onehas
Φ (119909) = 119909 (109)
The following result is the well-known Banach fixed pointtheorem Its proof can be found for example in [13Theorem348]
Theorem 45 (The Banach fixed point theorem) Let 119864 be acomplete metric space and let Φ 119864 rarr 119864 be a contractionThen the mapping Φ has a unique fixed point
In this text we will use the following slight extension ofthe Banach fixed point theorem
Corollary 46 Let 119864 be a complete metric space and let Φ
119864 rarr 119864 be a mapping such that for some 119899 isin N the mappingΦ119899 is a contraction Then the mapping Φ has a unique fixed
point
Proof According to the Banach fixed point theorem(Theorem 45) the mapping Φ
119899 has a unique fixed point thatis there exists a unique element 119909 isin 119864 such that Φ119899
(119909) = 119909Therefore we have
Φ (119909) = Φ (Φ119899
(119909)) = Φ119899
(Φ (119909)) (110)
showing thatΦ(119909) is a fixedpoint ofΦ119899 SinceΦ119899 has a uniquefixed point we deduce thatΦ(119909) = 119909 showing that119909 is a fixedpoint of Φ
In order to prove uniqueness let 119910 isin 119864 be another fixedpoint ofΦ that is we haveΦ(119910) = 119910 By induction we obtain
showing that 119910 is a fixed point of Φ119899 Since the mapping Φ119899
has exactly one fixed point we obtain 119909 = 119910
An indispensable tool for proving uniqueness of mildsolutions to (44) will be the following version of Gronwallrsquosinequality see for example [14 Theorem 51]
Lemma 47 (Gronwallrsquos inequality) Let 119879 ge 0 be fixed let119891 [0 119879] rarr R
and hence by the Cauchy-Schwarz inequality the Ito isom-etry (39) the growth estimate (9) from Lemma 10 and thelocal Lipschitz conditions (113) we obtain
Using Gronwallrsquos inequality (see Lemma 47) we deduce that119891 equiv 0 Thus by the continuity of the sample paths of 119883 and119884 we obtain
P(⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ) = 1 forall119899 isin N (120)
and hence by the continuity of the probability measureP weconclude that
P(⋂
119905ge0
119883119905and120591
1Γ= 119884
119905and1205911Γ)
= P(⋂
119899isinN
⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ)
= lim119899rarrinfin
P(⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ) = 1
(121)
which completes the proof
The local Lipschitz conditions (113) are in general notsufficient in order to ensure the existence of mild solutions tothe SPDE (44) Now we will prove that the existence of mildsolutions follows from global Lipschitz and linear growthconditions on 120572 and 120590 For this we recall an auxiliary resultwhich extends the Ito isometry (39)
Lemma 49 Let 119879 ge 0 be arbitrary and let 119883 = (119883119905)119905isin[0119879]
for all 119905 ge 0 and all ℎ isin 119867 Then for every F0-measurable
random variable ℎ0
Ω rarr 119867 there exists a (up toindistinguishability) unique mild solution 119883 to (44)
Proof The uniqueness of mild solutions to (44) is a directconsequence of Theorem 48 and hence we may concentrateon the existence proof which we divide into the followingseveral steps
Step 1 First we suppose that the initial condition ℎ0satisfies
E[ℎ02119901
] lt infin for some 119901 gt 1 Let 119879 ge 0 be arbitrary Wedefine the Banach space
1198712119901
119879(119867) = 119871
2119901
(Ω times [0 119879] P119879P otimes 119889119905119867) (129)
and prove that the variation of constants equation
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(130)
has a unique solution in the space 1198712119901
119879(119867) This is done in the
following three steps
Step 11 For 119883 isin 1198712119901
119879(119867) we define the process Φ119883 by
(Φ119883)119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(131)
Then the process Φ119883 is well defined Indeed by the growthestimate (9) the linear growth condition (127) and Holderrsquosinequality we have
Consequently there exists an index 119899 isin N such that Φ119899 is
a contraction and hence according to the extension of theBanach fixed point theorem (see Corollary 46) the mappingΦ has a unique fixed point 119883 isin 119871
2119901
119879(119867) This fixed point 119883 is
a solution to the variation of constants equation (130) Since119879 ge 0 was arbitrary there exists a process 119883 which is a solu-tion of the variation of constants equation
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
119905 ge 0
(146)
Step 14 In order to prove that 119883 is a mild solution to (44)it remains to show that 119883 has a continuous version ByLemma 12 the process
119905 997891997888rarr 119878119905ℎ0 119905 ge 0 (147)
is continuous and by Proposition 41 the process
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 119905 ge 0 (148)
is continuous too Moreover for every 119879 ge 0 we have by thelinear growth condition (128) Holderrsquos inequality and since119883 isin 119871
Thus by Proposition 43 the stochastic convolution 120590 ⋆ 119882
given by
(120590 ⋆ 119882)119905= int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0 (150)
has a continuous version and consequently the process119883hasa continuous version too This continuous version is a mildsolution to (44)
Step 2 Now let ℎ0 Ω rarr 119867 be an arbitraryF
0-measurable
random variable We define the sequence (ℎ119899)119899isinN of F
0-
measurable random variables as
ℎ119899
0= ℎ
01ℎ0le119899
119899 isin N (151)
Let 119899 isin N be arbitrary Then as ℎ119899
0is bounded we have
E[ℎ119899
02119901
] lt infin for all 119901 gt 1 By Step 1 the SPDE
119889119883119899
119905= (119860119883
119899
119905+ 120572 (119905 119883
119899
119905)) 119889119905 + 120590 (119905 119883
119899
119905) 119889119882
119905
119883119899
0= ℎ
119899
0
(152)
18 International Journal of Stochastic Analysis
has amild solution119883119899We define the sequence (Ω
119899)119899isinN sub F
0
as
Ω119899=
1003817100381710038171003817ℎ01003817100381710038171003817 le 119899 119899 isin N (153)
Then we have Ω119899sub Ω
119898for 119899 le 119898 Ω = ⋃
119899isinN Ω119899 and
Ω119899sub ℎ
119899
0= ℎ
119898
0 sub ℎ
119899
0= ℎ
0 forall119899 le 119898 (154)
Thus byTheorem 48 we have (up to indistinguishability)
119883119899
1Ω119899
= 119883119898
1Ω119899
forall119899 le 119898 (155)
Consequently the process
119883 = lim119899rarrinfin
119883119899
1Ω119899
(156)
is a well-defined continuous and adapted process and wehave
119883119899
1Ω119899
= 119883119898
1Ω119899
= 1198831Ω119899
forall119899 le 119898 (157)
Furthermore we obtain P-almost surely
119883119905= lim
119899rarrinfin
119883119899
1199051Ω119899
= lim119899rarrinfin
1Ω119899
(119878119905ℎ119899
0+ int
119905
0
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+int
119905
0
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ119899
0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= lim119899rarrinfin
1Ω119899
(119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= 119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(158)
proving that 119883 is a mild solution to (44)
Remark 51 For the proof of Theorem 50 we have usedCorollary 46 which is a slight extension of the Banach fixedpoint theorem Such an idea has been applied for example in[15]
Remark 52 A recent method for proving existence anduniqueness of mild solutions to the SPDE (44) is the methodof the moving frame presented in [6] see also [8] It allowsto reduce SPDE problems to the study of SDEs in infinitedimension In order to apply this method we need that thesemigroup (119878
119905)119905ge0
is a semigroup of pseudocontractions
We close this section with a consequence about theexistence of weak solutions
Corollary 53 Suppose that conditions (125)ndash(128) are ful-filled Let ℎ
0 Ω rarr 119867 be aF
0-measurable random variable
such that E[ℎ02119901
] lt infin for some 119901 gt 1 Then there exists a(up to indistinguishability) unique weak solution 119883 to (44)
Proof According to Proposition 35 every weak solution119883 to(44) is also a mild solution to (44)Therefore the uniquenessof weak solutions to (44) is a consequence of Theorem 48
It remains to prove the existence of a weak solution to(44) Let 119879 ge 0 be arbitrary By Theorem 50 and its proofthere exists a mild solution119883 isin 119871
2119901
119879(119867) to (44) By the linear
growth condition (128) and Holderrsquos inequality we obtain
showing that condition (72) is fulfilled Thus byProposition 37 the process 119883 is also a weak solution to(44)
8 Invariant Manifolds for WeakSolutions to SPDEs
In this section we deal with invariant manifolds for time-homogeneous SPDEs of the type (44) This topic arises fromthe natural desire to express the solutions of the SPDE (44)which generally live in the infinite dimensional Hilbert space119867 by means of a finite dimensional state process and thusto ensure larger analytical tractability Our goal is to findconditions on the generator 119860 and the coefficients 120572 120590 suchthat for every starting point of a finite dimensional submani-fold the solution process stays on the submanifold
We start with the required preliminaries about finitedimensional submanifolds inHilbert spaces In the sequel let119867 be a separable Hilbert space
Definition 54 Let 119898 119896 isin N be positive integers A subsetM sub 119867 is called an 119898-dimensional 119862119896-submanifold of 119867
International Journal of Stochastic Analysis 19
if for every ℎ isin M there exist an open neighborhood 119880 sub 119867
of ℎ an open set 119881 sub R119898 and a mapping 120601 isin 1198622
(119881119867) suchthat
(1) the mapping 120601 119881 rarr 119880 cap M is a homeomorphism(2) for all 119910 isin 119881 the mapping 119863120601(119910) is injective
The mapping 120601 is called a parametrization ofM around ℎ
In what follows let M be an 119898-dimensional 119862119896-
submanifold of 119867
Lemma 55 Let 120601119894
119881119894
rarr 119880119894cap M 119894 = 1 2 be two
parametrizations with119882= 1198801cap119880
2capM = 0Then themapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (160)
is a 119862119896-diffeomorphism
Proof See [16 Lemma 611]
Corollary 56 Let ℎ isin M be arbitrary and let 120601119894
119881119894
rarr
119880119894capM 119894 = 1 2 be two parametrizations ofM around ℎ Then
one has
1198631206011(119910
1) (R
119898
) = 1198631206012(119910
2) (R
119898
) (161)
where 119910119894= 120601
minus1
119894(ℎ) for 119894 = 1 2
Proof Since1198801and119880
2are open neighborhoods of ℎ we have
119882= 1198801cap 119880
2cap M = 0 Thus by Lemma 55 the mapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (162)
is a 119862119896-diffeomorphism Using the chain rule we obtain
1198631206012(119910
2) (R
119898
)
= 119863 (1206011∘ (120601
minus1
1∘ 120601
2)) (119910
2) (R
119898
)
= 1198631206011(119910
1)119863 (120601
minus1
1∘ 120601
2) (119910
2) (R
119898
)
sub 1198631206011(119910
1) (R
119898
)
(163)
and analogously we prove that 1198631206011(119910
1)(R119898
) sub
1198631206012(119910
2)(R119898
)
Definition 57 Let ℎ isin M be arbitraryThe tangent space ofMto ℎ is the subspace
119879ℎM = 119863120601 (119910) (R
119898
) (164)
where 119910 = 120601minus1
(ℎ) and 120601 119881 rarr 119880 cap M denotes aparametrization ofM around ℎ
Remark 58 Note that according to Corollary 56 theDefinition 57 of the tangent space 119879
ℎM does not depend on
the choice of the parametrization 120601 119881 rarr 119880 cap M
Proposition 59 Let ℎ isin M be arbitrary and let 120601 119881 rarr
119880 cap M be a parametrization ofM around ℎ Then there existan open set 119881
0sub 119881 an open neighborhood 119880
0sub 119880 of ℎ and
a mapping 120601 isin 119862119896
119887(R119898
119867) with 120601|1198810
= 120601|1198810
such that 120601|1198810
1198810
rarr 1198800cap M is a parametrization ofM around ℎ too
Proof See [16 Remark 611]
Remark 60 By Proposition 59 we may assume that anyparametrization 120601 119881 rarr 119880 cap M has an extension 120601 isin
119862119896
119887(R119898
119867)
Proposition 61 Let 119863 sub 119867 be a dense subset For every ℎ0isin
M there exist 1205771 120577
119898isin 119863 and a parametrization 120601 119881 rarr
119880 cap M around ℎ0such that
120601 (⟨120577 ℎ⟩) = ℎ forallℎ isin 119880 cap M (165)
where one uses the notation ⟨120577 ℎ⟩ = (⟨1205771 ℎ⟩ ⟨120577
119898 ℎ⟩) isin
R119898
Proof See [16 Proposition 612]
Proposition 62 Let 120601 119881 rarr 119880 cap M be a parametrizationas in Proposition 61 Then the following statements are true
(1) The elements 1205771 120577
119898are linearly independent in 119867
(2) For every ℎ isin 119880 cap M one has the direct sum decom-position
Therefore by Proposition 64 for every 1199100isin R119898 there exists
a (up to indistinguishability) unique weak solution to (192)which according to Proposition 39 is also a strong solutionto (192) The uniqueness of strong solutions to (192) is a con-sequence of Proposition 39 andTheorem 48
Now we are ready to formulate and prove ourmain resultof this section
Theorem 68 The following statements are equivalent
(1) The submanifoldM is locally invariant for (176)(2) One has
M sub D (119860) (195)
120590119895
(ℎ) isin 119879ℎM forallℎ isin M 119886119897119897 119895 isin N (196)
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) isin 119879ℎM forallℎ isin M (197)
(3) The operator 119860 is continuous onM and for each ℎ0isin
M there exists a local strong solution 119883 to (176) withsome lifetime 120591 gt 0 such that
119883119905and120591
isin M forall119905 ge 0 P-119886119897119898119900119904119905 119904119906119903119890119897119910 (198)
Proof (1)rArr(2) Let ℎ isin M be arbitrary By Proposition 61and Remark 60 there exist elements 120577
1 120577
119898isin D(119860
lowast
) anda parametrization 120601 119881 rarr 119880 cap M around ℎ such that theinverse 120601
minus1
119880 cap M rarr 119881 is given by 120601minus1
= ⟨120577 ∙⟩ and 120601
has an extension 120601 isin 1198622
119887(R119898
119867) Since the submanifoldM islocally invariant for (176) there exists a local weak solution119883
to (176) with initial condition ℎ and some lifetime 984858 gt 0 suchthat
119883119905and984858
isin M forall119905 ge 0 P-almost surely (199)
Since119880 is an open neighborhood of ℎ there exists 120598 gt 0 suchthat 119861
120598(ℎ) sub 119880 where 119861
120598(ℎ) denotes the open ball
119861120598(ℎ) = 119892 isin 119867
1003817100381710038171003817119892 minus ℎ1003817100381710038171003817 lt 120598 (200)
We define the stopping time
120591 = 984858 and inf 119905 ge 0 119883119905notin 119861
120598(ℎ) (201)
Since the process119883 has continuous sample paths and satisfies1198830= ℎ we have 120591 gt 0 and P-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (202)
22 International Journal of Stochastic Analysis
Defining the R119898-valued process 119884 = ⟨120577 119883⟩ we have P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (203)
Moreover since 119883 is a weak solution to (176) with initialcondition ℎ setting 119910 = ⟨120577 ℎ⟩ isin 119881 we have P-almost surely
119884119905and120591
= ⟨120577 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120577 120590119895
(119883119904)⟩ 119889120573
119895
119904
= ⟨120577 ℎ⟩ + int
119905and120591
0
120572120601120577
(⟨120577 119883119904⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(⟨120577 119883
119904⟩) 119889120573
119895
119904
= 119910 + int
119905and120591
0
120572120601120577
(119884119904) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(204)
showing that 119884 is a local strong solution to (192) with initialcondition 119910 By Itorsquos formula (Theorem 26) we obtain P-almost surely
119883119905and120591
= 120601 (119884119905and120591
)
= ℎ + int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904)))119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(205)
Now let 120585 isin D(119860lowast
) be arbitrary Then we have P-almostsurely
⟨120585 119883119905and120591
⟩
= ⟨120585 ℎ⟩
+ int
119905and120591
0
⟨120585119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 119863120601 (119884119904) 120590
119895
120601120577(119884
119904)⟩119889120573
119895
119904 119905 ge 0
(206)
On the other hand since 119883 is a local weak solution to (176)with initial condition ℎ and lifetime 120591 we have P-almostsurely for all 119905 ge 0 the identity
⟨120585 119883119905and120591
⟩ = ⟨120585 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904)⟩ 119889120573
119895
119904
(207)
Combining (206) and (207) yields up to indistinguishability
119861 + 119872 = 0 (208)
where the processes 119861 and 119872 are defined as
119861119905= int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩) 119889119904
119905 ge 0
119872119905= sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩ 119889120573
119895
119904
119905 ge 0
(209)
The process 119861 + 119872 is a continuous semimartingale withcanonical decomposition (208) Since the canonical decom-position of a continuous semimartingale is unique up toindistinguishability we deduce that 119861 = 119872 = 0 up toindistinguishability Using the Ito isometry (39) we obtain P-almost surely
int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩
+ ⟨120585 120572 (119883119904) minus 119863120601 (119884
showing that119883 is a local strong solution to (44) with lifetime120591
(3)rArr(1) This implication is a direct consequence ofProposition 29
The results from this section are closely related to the exis-tence of finite dimensional realizations that is the existenceof invariantmanifolds for each starting point ℎ
0 andwe point
out the papers [17ndash22] regarding this topic Furthermore wemention thatTheorem 68 has been extended in [23] to SPDEswith jumps
Acknowledgments
The author is grateful to Daniel Gaigall Georg GrafendorferFlorian Modler and Thomas Salfeld for the valuable com-ments and discussions
References
[1] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions CambridgeUniversity Press CambridgeUK 1992
[2] C Prevot and M Rockner A Concise Course on StochasticPartial Differential Equations Springer Berlin Germany 2007
[3] L Gawarecki and V Mandrekar Stochastic Differential Equa-tions in Infinite Dimensions with Applications to StochasticPartial Differential Equations Springer Berlin Germany 2011
[4] S Peszat and J Zabczyk Stochastic Partial Differential Equationswith Levy Noise Cambridge University Press Cambridge UK2007
[5] S Albeverio V Mandrekar and B Rudiger ldquoExistence of mildsolutions for stochastic differential equations and semilinearequations with non-Gaussian Levy noiserdquo Stochastic Processesand Their Applications vol 119 no 3 pp 835ndash863 2009
[6] D Filipovic S Tappe and J Teichmann ldquoJump-diffusions inHilbert spaces existence stability and numericsrdquo Stochasticsvol 82 no 5 pp 475ndash520 2010
[7] C Marinelli C Prevot and M Rockner ldquoRegular dependenceon initial data for stochastic evolution equations with multi-plicative Poisson noiserdquo Journal of Functional Analysis vol 258no 2 pp 616ndash649 2010
[8] S Tappe ldquoSome refinements of existence results for SPDEsdriven by Wiener processes and Poisson random measuresrdquoInternational Journal of Stochastic Analysis vol 2012 Article ID236327 24 pages 2012
[9] T Nakayama ldquoSupport theorem for mild solutions of SDErsquos inHilbert spacesrdquo Journal of Mathematical Sciences the Universityof Tokyo vol 11 no 3 pp 245ndash311 2004
[10] T Nakayama ldquoViability theorem for SPDErsquos including HJMframeworkrdquo Journal of Mathematical Sciences the University ofTokyo vol 11 no 3 pp 313ndash324 2004
[11] W Rudin Functional Analysis McGraw-Hill New York NYUSA 2nd edition 1991
[12] DWerner Funktionalanalysis Springer Berlin Germany 2007[13] C D Aliprantis and K C Border Infinite Dimensional Analysis
Springer Berlin Germany 3rd edition 2006[14] S N Ethier and T G KurtzMarkov Processes Characterization
andConvergenceWiley Series in Probability and Statistics JohnWiley amp Sons New Jersey NJ USA 1986
[15] V Mandrekar and B Rudiger ldquoExistence and uniqueness ofpath wise solutions for stochastic integral equations driven byLevy noise on separable Banach spacesrdquo Stochastics vol 78 no4 pp 189ndash212 2006
[16] D Filipovic Consistency Problems for Heath-Jarrow-MortonInterest Rate Models Springer Berlin Germany 2001
[17] T Bjork and L Svensson ldquoOn the existence of finite-dimensional realizations for nonlinear forward rate modelsrdquoMathematical Finance vol 11 no 2 pp 205ndash243 2001
[18] T Bjork and C Landen ldquoOn the construction of finite dimen-sional realizations for nonlinear forward rate modelsrdquo Financeand Stochastics vol 6 no 3 pp 303ndash331 2002
[19] D Filipovic and J Teichmann ldquoExistence of invariantmanifoldsfor stochastic equations in infinite dimensionrdquo Journal ofFunctional Analysis vol 197 no 2 pp 398ndash432 2003
[20] D Filipovic and J Teichmann ldquoOn the geometry of the termstructure of interest ratesrdquo Proceedings of The Royal Society ofLondon A vol 460 no 2041 pp 129ndash167 2004
[21] S Tappe ldquoAn alternative approach on the existence of affinerealizations for HJM term structure modelsrdquo Proceedings of
International Journal of Stochastic Analysis 25
TheRoyal Society of LondonA vol 466 no 2122 pp 3033ndash30602010
[22] S Tappe ldquoExistence of affine realizations for Levy term struc-ture modelsrdquo Proceedings of The Royal Society of London A vol468 no 2147 pp 3685ndash3704 2012
[23] D Filipovic S Tappe and J Teichmann ldquoInvariant manifoldswith boundary for jump-diffusionsrdquo httparxivorgabs12021076
constants 119872 ge 1 and 120596 isin R such that1003817100381710038171003817119878119905
1003817100381710038171003817 le 119872119890120596119905
forall119905 ge 0 (9)
Definition 11 Let (119878119905)119905ge0
be a 1198620-semigroup
(1) The semigroup (119878119905)119905ge0
is called a semigroup of contrac-tions (or contractive) if
10038171003817100381710038171198781199051003817100381710038171003817 le 1 forall119905 ge 0 (10)
that is the growth estimate (9) is satisfied with119872 = 1
and 120596 = 0(2) The semigroup (119878
119905)119905ge0
is called a semigroup of pseu-docontractions (or pseudocontractive) if there exists aconstant 120596 isin R such that
10038171003817100381710038171198781199051003817100381710038171003817 le 119890
120596119905
forall119905 ge 0 (11)
that is the growth estimate (9) is satisfiedwith119872 = 1
If (119878119905)119905ge0
is a semigroup of pseudocontractions withgrowth estimate (11) then (119879
119905)119905ge0
given by
119879119905= 119890
minus120596119905
119878119905 119905 ge 0 (12)
is a semigroup of contractions Hence every pseudocon-tractive semigroup can be transformed into a semigroup ofcontractions which explains the term pseudocontractive
(2) For all 119909 isin 119883 and 119905 ge 0 one has int1199050
119878119904119909119889119904 isin D(119860) and
119860(int
119905
0
119878119904119909 119889119904) = 119878
119905119909 minus 119909 (26)
(3) For all 119909 isin D(119860) and 119905 ge 0 one has
int
119905
0
119878119904119860119909119889119904 = 119878
119905119909 minus 119909 (27)
The following result shows that the strongly continuoussemigroup (119878
119905)119905ge0
associated with generator119860 is uniqueThisexplains the term generator
Proposition 19 Two 1198620-semigroups (119878
119905)119905ge0
and (119879119905)119905ge0
withthe same infinitesimal generator 119860 coincide that is one has119878119905= 119879
119905for all 119905 ge 0
The next result characterizes all norm continuous semi-groups in terms of their generators
Proposition 20 Let (119878119905)119905ge0
be a 1198620-semigroup with infinitesi-
mal generator119860 Then the following statements are equivalent
(1) The semigroup (119878119905)119905ge0
is norm continuous(2) The operator 119860 is continuous(3) The domain of 119860 is given byD(119860) = 119883
If the previous conditions are satisfied then one has 119878119905= 119890
119905119860
for all 119905 ge 0
Now we are interested in characterizing all linear opera-tors 119860 which are the infinitesimal generator of some stronglycontinuous semigroup (119878
119905)119905ge0
The following theorem ofHille-Yosida gives a characterization in terms of the resolventwhich we have introduced in Definition 3
Theorem 21 (Hille-Yosida theorem) Let 119860 119883 sup D(119860) rarr
119883 be a linear operator and let119872 ge 1120596 isin R be constantsThenthe following statements are equivalent
(1) 119860 is the generator of a 1198620-semigroup (119878
119905)119905ge0
withgrowth estimate (9)
(2) 119860 is densely defined and closed and one has (120596infin) sub
1198991003817100381710038171003817 le 119872(120582 minus 120596)minus119899
forall120582 isin (120596infin) 119899 isin N (28)
In particular we obtain the following characterization ofthe generators of semigroups of contractions
Corollary 22 For a linear operator 119860 119883 sup D(119860) rarr 119883 thefollowing statements are equivalent
(1) 119860 is the generator of a semigroup (119878119905)119905ge0
of contractions(2) 119860 is densely defined and closed and one has (0infin) sub
120588(119860) and
119877 (120582 119860) le1
120582forall120582 isin (0infin) (29)
Proposition 23 Let (119878119905)119905ge0
be a 1198620-semigroup on 119883 with
generator 119860 Then the family (119878119905|D(119860)
)119905ge0
is a 1198620-semigroup
on (D(119860) sdot D(119860)) with generator 119860 D(119860
2
) sub D(119860) rarr
D(1198602
) where the domain is given by
D (1198602
) = 119909 isin D (119860) 119860119909 isin D (119860) (30)
Recall that we have introduced the adjoint operator foroperators in the Hilbert spaces in Definition 6
Proposition 24 Let 119867 be a Hilbert space and let (119878119905)119905ge0
be a 1198620-semigroup on 119867 with generator 119860 Then the family
of adjoint operators (119878lowast
119905)119905ge0
is a 1198620-semigroup on 119867 with
generator 119860lowast
4 Stochastic Processes in Infinite Dimension
In this section we recall the required foundations aboutstochastic processes in infinite dimension In particular werecall the definition of a trace class Wiener process and out-line the construction of the Ito integral
In the sequel (ΩF (F119905)119905ge0
P) denotes a filtered prob-ability space satisfying the usual conditions Let H be aseparable Hilbert space and let 119876 isin 119871(H) be a nuclear self-adjoint positive definite linear operator
Definition 25 A H-valued adapted continuous process 119882
is called a 119876-Wiener process if the following conditions aresatisfied
(i) One has 1198820= 0
(ii) The random variable 119882119905minus 119882
119904and the 120590-algebra F
119904
are independent for all 0 le 119904 le 119905(iii) One has 119882
119905minus 119882
119904sim 119873(0 (119905 minus 119904)119876) for all 0 le 119904 le 119905
In Definition 25 the distribution 119873(0 (119905 minus 119904)119876) is aGaussian measure with mean 0 and covariance operator (119905 minus
119904)119876 see for example [1 Section 232]The operator119876 is alsocalled the covariance operator of the Wiener process 119882 As119876 is a trace class operator we also call 119882 a trace class Wienerprocess
Now let 119882 be a 119876-Wiener process Then there exist anorthonormal basis (119890
119895)119895isinN of H and a sequence (120582
119895)119895isinN sub
(0infin) with sum119895isinN 120582
119895lt infin such that
119876119906 = sum
119895isinN
120582119895⟨119906 119890
119895⟩H119890119895 119906 isin H (31)
International Journal of Stochastic Analysis 5
Namely the 120582119895are the eigenvalues of 119876 and each 119890
119895is an
eigenvector corresponding to 120582119895 The space H
0= 119876
12
(H)equipped with the inner product
⟨119906 V⟩H0
= ⟨119876minus12
119906 119876minus12V⟩
H (32)
is another separable Hilbert space and (radic120582119895119890119895)119895isinN
is anorthonormal basis According to [1 Proposition 41] thesequence of stochastic processes (120573119895)
119895isinN defined as
120573119895
=1
radic120582119895
⟨119882 119890119895⟩H 119895 isin N (33)
is a sequence of real-valued independent standard Wienerprocesses and one has the expansion
119882 = sum
119895isinN
radic120582119895120573119895
119890119895 (34)
Now let us briefly sketch the construction of the Ito integralwith respect to the Wiener process 119882 Further details can befound in [1 3] We denote by 119871
0
2(119867) = 119871
2(H
0 119867) the space
of Hilbert-Schmidt operators from H0into119867 endowed with
the Hilbert-Schmidt norm
Φ1198710
2(119867)
= (sum
119895isinN
120582119895
10038171003817100381710038171003817Φ119890
119895
10038171003817100381710038171003817
2
)
12
Φ isin 1198710
2(119867) (35)
which itself is a separable Hilbert space The construction ofthe Ito integral is divided into three steps as follows
(1) For every 119871(H 119867)-valued simple process of the form
119883 = 119883010
+
119899
sum
119894=1
1198831198941(119905119894119905119894+1] (36)
with 0 = 1199051
lt sdot sdot sdot lt 119905119899+1
= 119879 and F119905119894
-measurablerandom variables 119883
119894 Ω rarr 119871(H 119867) for 119894 = 1 119899
we set
int
119905
0
119883119904119889119882
119904=
119899
sum
119894=1
119883119894(119882
119905and119905119894+1
minus 119882119905and119905119894
) (37)
(2) For every predictable 1198710
2(119867)-valued process 119883 satis-
fying
E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904] lt infin (38)
we extend the Ito integral int1199050
119883119904119889119882
119904by an extension
argument for linear operators In particular we obtainthe Ito isometry as follows
) is dense in119867 and hencewe obtain P-almost surely
119883119905and120591
= ℎ0+ int
119905and120591
0
(119860119883119904+ 120572 (119904 119883
119904)) 119889119904
+ int
119905and120591
0
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(53)
Consequently the process 119883 is also a local strong solution to(44) with lifetime 120591
Corollary 31 Let M sub D(119860) be a subset such that 119860 iscontinuous on M and let 119883 be a local weak solution to (44)with lifetime 120591 such that
119883119905and120591
isin M forall119905 ge 0 P-119886119897119898119900119904119905 119904119906119903119890119897119910 (54)
Then 119883 is also a local strong solution to (44) with lifetime 120591
Proof Since M sub D(119860) condition (54) implies that (46) isfulfilled Moreover by the continuity of 119860 onM the samplepaths of the process 119860119883 are P-almost surely continuous andhence we obtain (47) Consequently using Proposition 30the process 119883 is also a local strong solution to (44) withlifetime 120591
Proposition 32 Every strong solution119883 to (44) is also a mildsolution to (44)
Proof According to Lemma 8 the domain (D(119860) sdot D(119860))
endowed with the graph norm is a separable Hilbert spacetoo Hence by Lemma 18 for all 119905 ge 0 the function
119891 [0 119905] times D (119860) 997888rarr 119867 119891 (119904 119909) = 119878119905minus119904
119909 (55)
belongs to the class 11986212loc119887
([0 119905] times D(119860)119867) with partialderivatives
Since by Proposition 14 the domain D((119860lowast
)2
) is dense in(D(119860
lowast
) sdot D(119860lowast)) we get P-almost surely for all 120577 isin D(119860
lowast
)
the identity
⟨120577 119883119905⟩ = ⟨120577 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904⟩
(70)
Since by Proposition 14 the domainD(119860lowast
) is dense in119867 weobtain P-almost surely
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 (71)
proving that 119883 is a mild solution to (44)
Remark 36 Now the proof of Proposition 32 is an immediateconsequence of Propositions 29 and 35
We have just seen that every weak solution to (44) is also amild solution Under the following regularity condition (72)the converse of this statement holds true as well
Proposition 37 Let 119883 be a mild solution to (44) such that
120572 (119904 119883119904) minus 120572 (119904 119883
119904)⟩ 119889119904
= ⟨120577 int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904⟩ minus int
119905
0
⟨120577 120572 (119904 119883119904)⟩119889119904
(74)
Due to assumption (72) we may use Fubinirsquos theorem forstochastic integrals (see [3 Theorem 28]) which togetherwith Lemma 18 gives us P-almost surely
Due to assumption (80) we may use Fubinirsquos theorem forstochastic integrals (see [3 Theorem 28]) which togetherwith Lemma 18 gives us P-almost surely
int
119905
0
(119878119905minus119904
120590 (119904 119883119904) minus 120590 (119904 119883
This proves that 119883 is also a strong solution to (44)
The following result shows that for norm continuoussemigroups the concepts of strong weak and mild solutionsare equivalent In particular this applies for finite dimen-sional state spaces
Proposition 39 Suppose that the semigroup (119878119905)119905ge0
is normcontinuous Let 119883 be a stochastic process with 119883
0= ℎ
0 Then
the following statements are equivalent
(1) The process 119883 is a strong solution to (44)
(2) The process 119883 is a weak solution to (44)
(3) The process 119883 is a mild solution to (44)
Proof (1)rArr(2) This implication is a consequence ofProposition 29
(2)rArr(3) This implication is a consequence ofProposition 35
(3)rArr(1) By Proposition 20 we have 119860 isin 119871(119867) and 119878119905=
119890119905119860 119905 ge 0 Furthermore the family (119890
119905119860
)119905isinR is a 119862
0-group on
119867 Therefore and since 119883 is a mild solution to (44) we haveP-almost surely
(2) An element 119909 isin 119864 is called a fixed point of Φ if onehas
Φ (119909) = 119909 (109)
The following result is the well-known Banach fixed pointtheorem Its proof can be found for example in [13Theorem348]
Theorem 45 (The Banach fixed point theorem) Let 119864 be acomplete metric space and let Φ 119864 rarr 119864 be a contractionThen the mapping Φ has a unique fixed point
In this text we will use the following slight extension ofthe Banach fixed point theorem
Corollary 46 Let 119864 be a complete metric space and let Φ
119864 rarr 119864 be a mapping such that for some 119899 isin N the mappingΦ119899 is a contraction Then the mapping Φ has a unique fixed
point
Proof According to the Banach fixed point theorem(Theorem 45) the mapping Φ
119899 has a unique fixed point thatis there exists a unique element 119909 isin 119864 such that Φ119899
(119909) = 119909Therefore we have
Φ (119909) = Φ (Φ119899
(119909)) = Φ119899
(Φ (119909)) (110)
showing thatΦ(119909) is a fixedpoint ofΦ119899 SinceΦ119899 has a uniquefixed point we deduce thatΦ(119909) = 119909 showing that119909 is a fixedpoint of Φ
In order to prove uniqueness let 119910 isin 119864 be another fixedpoint ofΦ that is we haveΦ(119910) = 119910 By induction we obtain
showing that 119910 is a fixed point of Φ119899 Since the mapping Φ119899
has exactly one fixed point we obtain 119909 = 119910
An indispensable tool for proving uniqueness of mildsolutions to (44) will be the following version of Gronwallrsquosinequality see for example [14 Theorem 51]
Lemma 47 (Gronwallrsquos inequality) Let 119879 ge 0 be fixed let119891 [0 119879] rarr R
and hence by the Cauchy-Schwarz inequality the Ito isom-etry (39) the growth estimate (9) from Lemma 10 and thelocal Lipschitz conditions (113) we obtain
Using Gronwallrsquos inequality (see Lemma 47) we deduce that119891 equiv 0 Thus by the continuity of the sample paths of 119883 and119884 we obtain
P(⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ) = 1 forall119899 isin N (120)
and hence by the continuity of the probability measureP weconclude that
P(⋂
119905ge0
119883119905and120591
1Γ= 119884
119905and1205911Γ)
= P(⋂
119899isinN
⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ)
= lim119899rarrinfin
P(⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ) = 1
(121)
which completes the proof
The local Lipschitz conditions (113) are in general notsufficient in order to ensure the existence of mild solutions tothe SPDE (44) Now we will prove that the existence of mildsolutions follows from global Lipschitz and linear growthconditions on 120572 and 120590 For this we recall an auxiliary resultwhich extends the Ito isometry (39)
Lemma 49 Let 119879 ge 0 be arbitrary and let 119883 = (119883119905)119905isin[0119879]
for all 119905 ge 0 and all ℎ isin 119867 Then for every F0-measurable
random variable ℎ0
Ω rarr 119867 there exists a (up toindistinguishability) unique mild solution 119883 to (44)
Proof The uniqueness of mild solutions to (44) is a directconsequence of Theorem 48 and hence we may concentrateon the existence proof which we divide into the followingseveral steps
Step 1 First we suppose that the initial condition ℎ0satisfies
E[ℎ02119901
] lt infin for some 119901 gt 1 Let 119879 ge 0 be arbitrary Wedefine the Banach space
1198712119901
119879(119867) = 119871
2119901
(Ω times [0 119879] P119879P otimes 119889119905119867) (129)
and prove that the variation of constants equation
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(130)
has a unique solution in the space 1198712119901
119879(119867) This is done in the
following three steps
Step 11 For 119883 isin 1198712119901
119879(119867) we define the process Φ119883 by
(Φ119883)119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(131)
Then the process Φ119883 is well defined Indeed by the growthestimate (9) the linear growth condition (127) and Holderrsquosinequality we have
Consequently there exists an index 119899 isin N such that Φ119899 is
a contraction and hence according to the extension of theBanach fixed point theorem (see Corollary 46) the mappingΦ has a unique fixed point 119883 isin 119871
2119901
119879(119867) This fixed point 119883 is
a solution to the variation of constants equation (130) Since119879 ge 0 was arbitrary there exists a process 119883 which is a solu-tion of the variation of constants equation
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
119905 ge 0
(146)
Step 14 In order to prove that 119883 is a mild solution to (44)it remains to show that 119883 has a continuous version ByLemma 12 the process
119905 997891997888rarr 119878119905ℎ0 119905 ge 0 (147)
is continuous and by Proposition 41 the process
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 119905 ge 0 (148)
is continuous too Moreover for every 119879 ge 0 we have by thelinear growth condition (128) Holderrsquos inequality and since119883 isin 119871
Thus by Proposition 43 the stochastic convolution 120590 ⋆ 119882
given by
(120590 ⋆ 119882)119905= int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0 (150)
has a continuous version and consequently the process119883hasa continuous version too This continuous version is a mildsolution to (44)
Step 2 Now let ℎ0 Ω rarr 119867 be an arbitraryF
0-measurable
random variable We define the sequence (ℎ119899)119899isinN of F
0-
measurable random variables as
ℎ119899
0= ℎ
01ℎ0le119899
119899 isin N (151)
Let 119899 isin N be arbitrary Then as ℎ119899
0is bounded we have
E[ℎ119899
02119901
] lt infin for all 119901 gt 1 By Step 1 the SPDE
119889119883119899
119905= (119860119883
119899
119905+ 120572 (119905 119883
119899
119905)) 119889119905 + 120590 (119905 119883
119899
119905) 119889119882
119905
119883119899
0= ℎ
119899
0
(152)
18 International Journal of Stochastic Analysis
has amild solution119883119899We define the sequence (Ω
119899)119899isinN sub F
0
as
Ω119899=
1003817100381710038171003817ℎ01003817100381710038171003817 le 119899 119899 isin N (153)
Then we have Ω119899sub Ω
119898for 119899 le 119898 Ω = ⋃
119899isinN Ω119899 and
Ω119899sub ℎ
119899
0= ℎ
119898
0 sub ℎ
119899
0= ℎ
0 forall119899 le 119898 (154)
Thus byTheorem 48 we have (up to indistinguishability)
119883119899
1Ω119899
= 119883119898
1Ω119899
forall119899 le 119898 (155)
Consequently the process
119883 = lim119899rarrinfin
119883119899
1Ω119899
(156)
is a well-defined continuous and adapted process and wehave
119883119899
1Ω119899
= 119883119898
1Ω119899
= 1198831Ω119899
forall119899 le 119898 (157)
Furthermore we obtain P-almost surely
119883119905= lim
119899rarrinfin
119883119899
1199051Ω119899
= lim119899rarrinfin
1Ω119899
(119878119905ℎ119899
0+ int
119905
0
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+int
119905
0
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ119899
0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= lim119899rarrinfin
1Ω119899
(119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= 119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(158)
proving that 119883 is a mild solution to (44)
Remark 51 For the proof of Theorem 50 we have usedCorollary 46 which is a slight extension of the Banach fixedpoint theorem Such an idea has been applied for example in[15]
Remark 52 A recent method for proving existence anduniqueness of mild solutions to the SPDE (44) is the methodof the moving frame presented in [6] see also [8] It allowsto reduce SPDE problems to the study of SDEs in infinitedimension In order to apply this method we need that thesemigroup (119878
119905)119905ge0
is a semigroup of pseudocontractions
We close this section with a consequence about theexistence of weak solutions
Corollary 53 Suppose that conditions (125)ndash(128) are ful-filled Let ℎ
0 Ω rarr 119867 be aF
0-measurable random variable
such that E[ℎ02119901
] lt infin for some 119901 gt 1 Then there exists a(up to indistinguishability) unique weak solution 119883 to (44)
Proof According to Proposition 35 every weak solution119883 to(44) is also a mild solution to (44)Therefore the uniquenessof weak solutions to (44) is a consequence of Theorem 48
It remains to prove the existence of a weak solution to(44) Let 119879 ge 0 be arbitrary By Theorem 50 and its proofthere exists a mild solution119883 isin 119871
2119901
119879(119867) to (44) By the linear
growth condition (128) and Holderrsquos inequality we obtain
showing that condition (72) is fulfilled Thus byProposition 37 the process 119883 is also a weak solution to(44)
8 Invariant Manifolds for WeakSolutions to SPDEs
In this section we deal with invariant manifolds for time-homogeneous SPDEs of the type (44) This topic arises fromthe natural desire to express the solutions of the SPDE (44)which generally live in the infinite dimensional Hilbert space119867 by means of a finite dimensional state process and thusto ensure larger analytical tractability Our goal is to findconditions on the generator 119860 and the coefficients 120572 120590 suchthat for every starting point of a finite dimensional submani-fold the solution process stays on the submanifold
We start with the required preliminaries about finitedimensional submanifolds inHilbert spaces In the sequel let119867 be a separable Hilbert space
Definition 54 Let 119898 119896 isin N be positive integers A subsetM sub 119867 is called an 119898-dimensional 119862119896-submanifold of 119867
International Journal of Stochastic Analysis 19
if for every ℎ isin M there exist an open neighborhood 119880 sub 119867
of ℎ an open set 119881 sub R119898 and a mapping 120601 isin 1198622
(119881119867) suchthat
(1) the mapping 120601 119881 rarr 119880 cap M is a homeomorphism(2) for all 119910 isin 119881 the mapping 119863120601(119910) is injective
The mapping 120601 is called a parametrization ofM around ℎ
In what follows let M be an 119898-dimensional 119862119896-
submanifold of 119867
Lemma 55 Let 120601119894
119881119894
rarr 119880119894cap M 119894 = 1 2 be two
parametrizations with119882= 1198801cap119880
2capM = 0Then themapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (160)
is a 119862119896-diffeomorphism
Proof See [16 Lemma 611]
Corollary 56 Let ℎ isin M be arbitrary and let 120601119894
119881119894
rarr
119880119894capM 119894 = 1 2 be two parametrizations ofM around ℎ Then
one has
1198631206011(119910
1) (R
119898
) = 1198631206012(119910
2) (R
119898
) (161)
where 119910119894= 120601
minus1
119894(ℎ) for 119894 = 1 2
Proof Since1198801and119880
2are open neighborhoods of ℎ we have
119882= 1198801cap 119880
2cap M = 0 Thus by Lemma 55 the mapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (162)
is a 119862119896-diffeomorphism Using the chain rule we obtain
1198631206012(119910
2) (R
119898
)
= 119863 (1206011∘ (120601
minus1
1∘ 120601
2)) (119910
2) (R
119898
)
= 1198631206011(119910
1)119863 (120601
minus1
1∘ 120601
2) (119910
2) (R
119898
)
sub 1198631206011(119910
1) (R
119898
)
(163)
and analogously we prove that 1198631206011(119910
1)(R119898
) sub
1198631206012(119910
2)(R119898
)
Definition 57 Let ℎ isin M be arbitraryThe tangent space ofMto ℎ is the subspace
119879ℎM = 119863120601 (119910) (R
119898
) (164)
where 119910 = 120601minus1
(ℎ) and 120601 119881 rarr 119880 cap M denotes aparametrization ofM around ℎ
Remark 58 Note that according to Corollary 56 theDefinition 57 of the tangent space 119879
ℎM does not depend on
the choice of the parametrization 120601 119881 rarr 119880 cap M
Proposition 59 Let ℎ isin M be arbitrary and let 120601 119881 rarr
119880 cap M be a parametrization ofM around ℎ Then there existan open set 119881
0sub 119881 an open neighborhood 119880
0sub 119880 of ℎ and
a mapping 120601 isin 119862119896
119887(R119898
119867) with 120601|1198810
= 120601|1198810
such that 120601|1198810
1198810
rarr 1198800cap M is a parametrization ofM around ℎ too
Proof See [16 Remark 611]
Remark 60 By Proposition 59 we may assume that anyparametrization 120601 119881 rarr 119880 cap M has an extension 120601 isin
119862119896
119887(R119898
119867)
Proposition 61 Let 119863 sub 119867 be a dense subset For every ℎ0isin
M there exist 1205771 120577
119898isin 119863 and a parametrization 120601 119881 rarr
119880 cap M around ℎ0such that
120601 (⟨120577 ℎ⟩) = ℎ forallℎ isin 119880 cap M (165)
where one uses the notation ⟨120577 ℎ⟩ = (⟨1205771 ℎ⟩ ⟨120577
119898 ℎ⟩) isin
R119898
Proof See [16 Proposition 612]
Proposition 62 Let 120601 119881 rarr 119880 cap M be a parametrizationas in Proposition 61 Then the following statements are true
(1) The elements 1205771 120577
119898are linearly independent in 119867
(2) For every ℎ isin 119880 cap M one has the direct sum decom-position
Therefore by Proposition 64 for every 1199100isin R119898 there exists
a (up to indistinguishability) unique weak solution to (192)which according to Proposition 39 is also a strong solutionto (192) The uniqueness of strong solutions to (192) is a con-sequence of Proposition 39 andTheorem 48
Now we are ready to formulate and prove ourmain resultof this section
Theorem 68 The following statements are equivalent
(1) The submanifoldM is locally invariant for (176)(2) One has
M sub D (119860) (195)
120590119895
(ℎ) isin 119879ℎM forallℎ isin M 119886119897119897 119895 isin N (196)
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) isin 119879ℎM forallℎ isin M (197)
(3) The operator 119860 is continuous onM and for each ℎ0isin
M there exists a local strong solution 119883 to (176) withsome lifetime 120591 gt 0 such that
119883119905and120591
isin M forall119905 ge 0 P-119886119897119898119900119904119905 119904119906119903119890119897119910 (198)
Proof (1)rArr(2) Let ℎ isin M be arbitrary By Proposition 61and Remark 60 there exist elements 120577
1 120577
119898isin D(119860
lowast
) anda parametrization 120601 119881 rarr 119880 cap M around ℎ such that theinverse 120601
minus1
119880 cap M rarr 119881 is given by 120601minus1
= ⟨120577 ∙⟩ and 120601
has an extension 120601 isin 1198622
119887(R119898
119867) Since the submanifoldM islocally invariant for (176) there exists a local weak solution119883
to (176) with initial condition ℎ and some lifetime 984858 gt 0 suchthat
119883119905and984858
isin M forall119905 ge 0 P-almost surely (199)
Since119880 is an open neighborhood of ℎ there exists 120598 gt 0 suchthat 119861
120598(ℎ) sub 119880 where 119861
120598(ℎ) denotes the open ball
119861120598(ℎ) = 119892 isin 119867
1003817100381710038171003817119892 minus ℎ1003817100381710038171003817 lt 120598 (200)
We define the stopping time
120591 = 984858 and inf 119905 ge 0 119883119905notin 119861
120598(ℎ) (201)
Since the process119883 has continuous sample paths and satisfies1198830= ℎ we have 120591 gt 0 and P-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (202)
22 International Journal of Stochastic Analysis
Defining the R119898-valued process 119884 = ⟨120577 119883⟩ we have P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (203)
Moreover since 119883 is a weak solution to (176) with initialcondition ℎ setting 119910 = ⟨120577 ℎ⟩ isin 119881 we have P-almost surely
119884119905and120591
= ⟨120577 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120577 120590119895
(119883119904)⟩ 119889120573
119895
119904
= ⟨120577 ℎ⟩ + int
119905and120591
0
120572120601120577
(⟨120577 119883119904⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(⟨120577 119883
119904⟩) 119889120573
119895
119904
= 119910 + int
119905and120591
0
120572120601120577
(119884119904) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(204)
showing that 119884 is a local strong solution to (192) with initialcondition 119910 By Itorsquos formula (Theorem 26) we obtain P-almost surely
119883119905and120591
= 120601 (119884119905and120591
)
= ℎ + int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904)))119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(205)
Now let 120585 isin D(119860lowast
) be arbitrary Then we have P-almostsurely
⟨120585 119883119905and120591
⟩
= ⟨120585 ℎ⟩
+ int
119905and120591
0
⟨120585119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 119863120601 (119884119904) 120590
119895
120601120577(119884
119904)⟩119889120573
119895
119904 119905 ge 0
(206)
On the other hand since 119883 is a local weak solution to (176)with initial condition ℎ and lifetime 120591 we have P-almostsurely for all 119905 ge 0 the identity
⟨120585 119883119905and120591
⟩ = ⟨120585 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904)⟩ 119889120573
119895
119904
(207)
Combining (206) and (207) yields up to indistinguishability
119861 + 119872 = 0 (208)
where the processes 119861 and 119872 are defined as
119861119905= int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩) 119889119904
119905 ge 0
119872119905= sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩ 119889120573
119895
119904
119905 ge 0
(209)
The process 119861 + 119872 is a continuous semimartingale withcanonical decomposition (208) Since the canonical decom-position of a continuous semimartingale is unique up toindistinguishability we deduce that 119861 = 119872 = 0 up toindistinguishability Using the Ito isometry (39) we obtain P-almost surely
int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩
+ ⟨120585 120572 (119883119904) minus 119863120601 (119884
showing that119883 is a local strong solution to (44) with lifetime120591
(3)rArr(1) This implication is a direct consequence ofProposition 29
The results from this section are closely related to the exis-tence of finite dimensional realizations that is the existenceof invariantmanifolds for each starting point ℎ
0 andwe point
out the papers [17ndash22] regarding this topic Furthermore wemention thatTheorem 68 has been extended in [23] to SPDEswith jumps
Acknowledgments
The author is grateful to Daniel Gaigall Georg GrafendorferFlorian Modler and Thomas Salfeld for the valuable com-ments and discussions
References
[1] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions CambridgeUniversity Press CambridgeUK 1992
[2] C Prevot and M Rockner A Concise Course on StochasticPartial Differential Equations Springer Berlin Germany 2007
[3] L Gawarecki and V Mandrekar Stochastic Differential Equa-tions in Infinite Dimensions with Applications to StochasticPartial Differential Equations Springer Berlin Germany 2011
[4] S Peszat and J Zabczyk Stochastic Partial Differential Equationswith Levy Noise Cambridge University Press Cambridge UK2007
[5] S Albeverio V Mandrekar and B Rudiger ldquoExistence of mildsolutions for stochastic differential equations and semilinearequations with non-Gaussian Levy noiserdquo Stochastic Processesand Their Applications vol 119 no 3 pp 835ndash863 2009
[6] D Filipovic S Tappe and J Teichmann ldquoJump-diffusions inHilbert spaces existence stability and numericsrdquo Stochasticsvol 82 no 5 pp 475ndash520 2010
[7] C Marinelli C Prevot and M Rockner ldquoRegular dependenceon initial data for stochastic evolution equations with multi-plicative Poisson noiserdquo Journal of Functional Analysis vol 258no 2 pp 616ndash649 2010
[8] S Tappe ldquoSome refinements of existence results for SPDEsdriven by Wiener processes and Poisson random measuresrdquoInternational Journal of Stochastic Analysis vol 2012 Article ID236327 24 pages 2012
[9] T Nakayama ldquoSupport theorem for mild solutions of SDErsquos inHilbert spacesrdquo Journal of Mathematical Sciences the Universityof Tokyo vol 11 no 3 pp 245ndash311 2004
[10] T Nakayama ldquoViability theorem for SPDErsquos including HJMframeworkrdquo Journal of Mathematical Sciences the University ofTokyo vol 11 no 3 pp 313ndash324 2004
[11] W Rudin Functional Analysis McGraw-Hill New York NYUSA 2nd edition 1991
[12] DWerner Funktionalanalysis Springer Berlin Germany 2007[13] C D Aliprantis and K C Border Infinite Dimensional Analysis
Springer Berlin Germany 3rd edition 2006[14] S N Ethier and T G KurtzMarkov Processes Characterization
andConvergenceWiley Series in Probability and Statistics JohnWiley amp Sons New Jersey NJ USA 1986
[15] V Mandrekar and B Rudiger ldquoExistence and uniqueness ofpath wise solutions for stochastic integral equations driven byLevy noise on separable Banach spacesrdquo Stochastics vol 78 no4 pp 189ndash212 2006
[16] D Filipovic Consistency Problems for Heath-Jarrow-MortonInterest Rate Models Springer Berlin Germany 2001
[17] T Bjork and L Svensson ldquoOn the existence of finite-dimensional realizations for nonlinear forward rate modelsrdquoMathematical Finance vol 11 no 2 pp 205ndash243 2001
[18] T Bjork and C Landen ldquoOn the construction of finite dimen-sional realizations for nonlinear forward rate modelsrdquo Financeand Stochastics vol 6 no 3 pp 303ndash331 2002
[19] D Filipovic and J Teichmann ldquoExistence of invariantmanifoldsfor stochastic equations in infinite dimensionrdquo Journal ofFunctional Analysis vol 197 no 2 pp 398ndash432 2003
[20] D Filipovic and J Teichmann ldquoOn the geometry of the termstructure of interest ratesrdquo Proceedings of The Royal Society ofLondon A vol 460 no 2041 pp 129ndash167 2004
[21] S Tappe ldquoAn alternative approach on the existence of affinerealizations for HJM term structure modelsrdquo Proceedings of
International Journal of Stochastic Analysis 25
TheRoyal Society of LondonA vol 466 no 2122 pp 3033ndash30602010
[22] S Tappe ldquoExistence of affine realizations for Levy term struc-ture modelsrdquo Proceedings of The Royal Society of London A vol468 no 2147 pp 3685ndash3704 2012
[23] D Filipovic S Tappe and J Teichmann ldquoInvariant manifoldswith boundary for jump-diffusionsrdquo httparxivorgabs12021076
(2) For all 119909 isin 119883 and 119905 ge 0 one has int1199050
119878119904119909119889119904 isin D(119860) and
119860(int
119905
0
119878119904119909 119889119904) = 119878
119905119909 minus 119909 (26)
(3) For all 119909 isin D(119860) and 119905 ge 0 one has
int
119905
0
119878119904119860119909119889119904 = 119878
119905119909 minus 119909 (27)
The following result shows that the strongly continuoussemigroup (119878
119905)119905ge0
associated with generator119860 is uniqueThisexplains the term generator
Proposition 19 Two 1198620-semigroups (119878
119905)119905ge0
and (119879119905)119905ge0
withthe same infinitesimal generator 119860 coincide that is one has119878119905= 119879
119905for all 119905 ge 0
The next result characterizes all norm continuous semi-groups in terms of their generators
Proposition 20 Let (119878119905)119905ge0
be a 1198620-semigroup with infinitesi-
mal generator119860 Then the following statements are equivalent
(1) The semigroup (119878119905)119905ge0
is norm continuous(2) The operator 119860 is continuous(3) The domain of 119860 is given byD(119860) = 119883
If the previous conditions are satisfied then one has 119878119905= 119890
119905119860
for all 119905 ge 0
Now we are interested in characterizing all linear opera-tors 119860 which are the infinitesimal generator of some stronglycontinuous semigroup (119878
119905)119905ge0
The following theorem ofHille-Yosida gives a characterization in terms of the resolventwhich we have introduced in Definition 3
Theorem 21 (Hille-Yosida theorem) Let 119860 119883 sup D(119860) rarr
119883 be a linear operator and let119872 ge 1120596 isin R be constantsThenthe following statements are equivalent
(1) 119860 is the generator of a 1198620-semigroup (119878
119905)119905ge0
withgrowth estimate (9)
(2) 119860 is densely defined and closed and one has (120596infin) sub
1198991003817100381710038171003817 le 119872(120582 minus 120596)minus119899
forall120582 isin (120596infin) 119899 isin N (28)
In particular we obtain the following characterization ofthe generators of semigroups of contractions
Corollary 22 For a linear operator 119860 119883 sup D(119860) rarr 119883 thefollowing statements are equivalent
(1) 119860 is the generator of a semigroup (119878119905)119905ge0
of contractions(2) 119860 is densely defined and closed and one has (0infin) sub
120588(119860) and
119877 (120582 119860) le1
120582forall120582 isin (0infin) (29)
Proposition 23 Let (119878119905)119905ge0
be a 1198620-semigroup on 119883 with
generator 119860 Then the family (119878119905|D(119860)
)119905ge0
is a 1198620-semigroup
on (D(119860) sdot D(119860)) with generator 119860 D(119860
2
) sub D(119860) rarr
D(1198602
) where the domain is given by
D (1198602
) = 119909 isin D (119860) 119860119909 isin D (119860) (30)
Recall that we have introduced the adjoint operator foroperators in the Hilbert spaces in Definition 6
Proposition 24 Let 119867 be a Hilbert space and let (119878119905)119905ge0
be a 1198620-semigroup on 119867 with generator 119860 Then the family
of adjoint operators (119878lowast
119905)119905ge0
is a 1198620-semigroup on 119867 with
generator 119860lowast
4 Stochastic Processes in Infinite Dimension
In this section we recall the required foundations aboutstochastic processes in infinite dimension In particular werecall the definition of a trace class Wiener process and out-line the construction of the Ito integral
In the sequel (ΩF (F119905)119905ge0
P) denotes a filtered prob-ability space satisfying the usual conditions Let H be aseparable Hilbert space and let 119876 isin 119871(H) be a nuclear self-adjoint positive definite linear operator
Definition 25 A H-valued adapted continuous process 119882
is called a 119876-Wiener process if the following conditions aresatisfied
(i) One has 1198820= 0
(ii) The random variable 119882119905minus 119882
119904and the 120590-algebra F
119904
are independent for all 0 le 119904 le 119905(iii) One has 119882
119905minus 119882
119904sim 119873(0 (119905 minus 119904)119876) for all 0 le 119904 le 119905
In Definition 25 the distribution 119873(0 (119905 minus 119904)119876) is aGaussian measure with mean 0 and covariance operator (119905 minus
119904)119876 see for example [1 Section 232]The operator119876 is alsocalled the covariance operator of the Wiener process 119882 As119876 is a trace class operator we also call 119882 a trace class Wienerprocess
Now let 119882 be a 119876-Wiener process Then there exist anorthonormal basis (119890
119895)119895isinN of H and a sequence (120582
119895)119895isinN sub
(0infin) with sum119895isinN 120582
119895lt infin such that
119876119906 = sum
119895isinN
120582119895⟨119906 119890
119895⟩H119890119895 119906 isin H (31)
International Journal of Stochastic Analysis 5
Namely the 120582119895are the eigenvalues of 119876 and each 119890
119895is an
eigenvector corresponding to 120582119895 The space H
0= 119876
12
(H)equipped with the inner product
⟨119906 V⟩H0
= ⟨119876minus12
119906 119876minus12V⟩
H (32)
is another separable Hilbert space and (radic120582119895119890119895)119895isinN
is anorthonormal basis According to [1 Proposition 41] thesequence of stochastic processes (120573119895)
119895isinN defined as
120573119895
=1
radic120582119895
⟨119882 119890119895⟩H 119895 isin N (33)
is a sequence of real-valued independent standard Wienerprocesses and one has the expansion
119882 = sum
119895isinN
radic120582119895120573119895
119890119895 (34)
Now let us briefly sketch the construction of the Ito integralwith respect to the Wiener process 119882 Further details can befound in [1 3] We denote by 119871
0
2(119867) = 119871
2(H
0 119867) the space
of Hilbert-Schmidt operators from H0into119867 endowed with
the Hilbert-Schmidt norm
Φ1198710
2(119867)
= (sum
119895isinN
120582119895
10038171003817100381710038171003817Φ119890
119895
10038171003817100381710038171003817
2
)
12
Φ isin 1198710
2(119867) (35)
which itself is a separable Hilbert space The construction ofthe Ito integral is divided into three steps as follows
(1) For every 119871(H 119867)-valued simple process of the form
119883 = 119883010
+
119899
sum
119894=1
1198831198941(119905119894119905119894+1] (36)
with 0 = 1199051
lt sdot sdot sdot lt 119905119899+1
= 119879 and F119905119894
-measurablerandom variables 119883
119894 Ω rarr 119871(H 119867) for 119894 = 1 119899
we set
int
119905
0
119883119904119889119882
119904=
119899
sum
119894=1
119883119894(119882
119905and119905119894+1
minus 119882119905and119905119894
) (37)
(2) For every predictable 1198710
2(119867)-valued process 119883 satis-
fying
E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904] lt infin (38)
we extend the Ito integral int1199050
119883119904119889119882
119904by an extension
argument for linear operators In particular we obtainthe Ito isometry as follows
) is dense in119867 and hencewe obtain P-almost surely
119883119905and120591
= ℎ0+ int
119905and120591
0
(119860119883119904+ 120572 (119904 119883
119904)) 119889119904
+ int
119905and120591
0
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(53)
Consequently the process 119883 is also a local strong solution to(44) with lifetime 120591
Corollary 31 Let M sub D(119860) be a subset such that 119860 iscontinuous on M and let 119883 be a local weak solution to (44)with lifetime 120591 such that
119883119905and120591
isin M forall119905 ge 0 P-119886119897119898119900119904119905 119904119906119903119890119897119910 (54)
Then 119883 is also a local strong solution to (44) with lifetime 120591
Proof Since M sub D(119860) condition (54) implies that (46) isfulfilled Moreover by the continuity of 119860 onM the samplepaths of the process 119860119883 are P-almost surely continuous andhence we obtain (47) Consequently using Proposition 30the process 119883 is also a local strong solution to (44) withlifetime 120591
Proposition 32 Every strong solution119883 to (44) is also a mildsolution to (44)
Proof According to Lemma 8 the domain (D(119860) sdot D(119860))
endowed with the graph norm is a separable Hilbert spacetoo Hence by Lemma 18 for all 119905 ge 0 the function
119891 [0 119905] times D (119860) 997888rarr 119867 119891 (119904 119909) = 119878119905minus119904
119909 (55)
belongs to the class 11986212loc119887
([0 119905] times D(119860)119867) with partialderivatives
Since by Proposition 14 the domain D((119860lowast
)2
) is dense in(D(119860
lowast
) sdot D(119860lowast)) we get P-almost surely for all 120577 isin D(119860
lowast
)
the identity
⟨120577 119883119905⟩ = ⟨120577 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904⟩
(70)
Since by Proposition 14 the domainD(119860lowast
) is dense in119867 weobtain P-almost surely
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 (71)
proving that 119883 is a mild solution to (44)
Remark 36 Now the proof of Proposition 32 is an immediateconsequence of Propositions 29 and 35
We have just seen that every weak solution to (44) is also amild solution Under the following regularity condition (72)the converse of this statement holds true as well
Proposition 37 Let 119883 be a mild solution to (44) such that
120572 (119904 119883119904) minus 120572 (119904 119883
119904)⟩ 119889119904
= ⟨120577 int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904⟩ minus int
119905
0
⟨120577 120572 (119904 119883119904)⟩119889119904
(74)
Due to assumption (72) we may use Fubinirsquos theorem forstochastic integrals (see [3 Theorem 28]) which togetherwith Lemma 18 gives us P-almost surely
Due to assumption (80) we may use Fubinirsquos theorem forstochastic integrals (see [3 Theorem 28]) which togetherwith Lemma 18 gives us P-almost surely
int
119905
0
(119878119905minus119904
120590 (119904 119883119904) minus 120590 (119904 119883
This proves that 119883 is also a strong solution to (44)
The following result shows that for norm continuoussemigroups the concepts of strong weak and mild solutionsare equivalent In particular this applies for finite dimen-sional state spaces
Proposition 39 Suppose that the semigroup (119878119905)119905ge0
is normcontinuous Let 119883 be a stochastic process with 119883
0= ℎ
0 Then
the following statements are equivalent
(1) The process 119883 is a strong solution to (44)
(2) The process 119883 is a weak solution to (44)
(3) The process 119883 is a mild solution to (44)
Proof (1)rArr(2) This implication is a consequence ofProposition 29
(2)rArr(3) This implication is a consequence ofProposition 35
(3)rArr(1) By Proposition 20 we have 119860 isin 119871(119867) and 119878119905=
119890119905119860 119905 ge 0 Furthermore the family (119890
119905119860
)119905isinR is a 119862
0-group on
119867 Therefore and since 119883 is a mild solution to (44) we haveP-almost surely
(2) An element 119909 isin 119864 is called a fixed point of Φ if onehas
Φ (119909) = 119909 (109)
The following result is the well-known Banach fixed pointtheorem Its proof can be found for example in [13Theorem348]
Theorem 45 (The Banach fixed point theorem) Let 119864 be acomplete metric space and let Φ 119864 rarr 119864 be a contractionThen the mapping Φ has a unique fixed point
In this text we will use the following slight extension ofthe Banach fixed point theorem
Corollary 46 Let 119864 be a complete metric space and let Φ
119864 rarr 119864 be a mapping such that for some 119899 isin N the mappingΦ119899 is a contraction Then the mapping Φ has a unique fixed
point
Proof According to the Banach fixed point theorem(Theorem 45) the mapping Φ
119899 has a unique fixed point thatis there exists a unique element 119909 isin 119864 such that Φ119899
(119909) = 119909Therefore we have
Φ (119909) = Φ (Φ119899
(119909)) = Φ119899
(Φ (119909)) (110)
showing thatΦ(119909) is a fixedpoint ofΦ119899 SinceΦ119899 has a uniquefixed point we deduce thatΦ(119909) = 119909 showing that119909 is a fixedpoint of Φ
In order to prove uniqueness let 119910 isin 119864 be another fixedpoint ofΦ that is we haveΦ(119910) = 119910 By induction we obtain
showing that 119910 is a fixed point of Φ119899 Since the mapping Φ119899
has exactly one fixed point we obtain 119909 = 119910
An indispensable tool for proving uniqueness of mildsolutions to (44) will be the following version of Gronwallrsquosinequality see for example [14 Theorem 51]
Lemma 47 (Gronwallrsquos inequality) Let 119879 ge 0 be fixed let119891 [0 119879] rarr R
and hence by the Cauchy-Schwarz inequality the Ito isom-etry (39) the growth estimate (9) from Lemma 10 and thelocal Lipschitz conditions (113) we obtain
Using Gronwallrsquos inequality (see Lemma 47) we deduce that119891 equiv 0 Thus by the continuity of the sample paths of 119883 and119884 we obtain
P(⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ) = 1 forall119899 isin N (120)
and hence by the continuity of the probability measureP weconclude that
P(⋂
119905ge0
119883119905and120591
1Γ= 119884
119905and1205911Γ)
= P(⋂
119899isinN
⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ)
= lim119899rarrinfin
P(⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ) = 1
(121)
which completes the proof
The local Lipschitz conditions (113) are in general notsufficient in order to ensure the existence of mild solutions tothe SPDE (44) Now we will prove that the existence of mildsolutions follows from global Lipschitz and linear growthconditions on 120572 and 120590 For this we recall an auxiliary resultwhich extends the Ito isometry (39)
Lemma 49 Let 119879 ge 0 be arbitrary and let 119883 = (119883119905)119905isin[0119879]
for all 119905 ge 0 and all ℎ isin 119867 Then for every F0-measurable
random variable ℎ0
Ω rarr 119867 there exists a (up toindistinguishability) unique mild solution 119883 to (44)
Proof The uniqueness of mild solutions to (44) is a directconsequence of Theorem 48 and hence we may concentrateon the existence proof which we divide into the followingseveral steps
Step 1 First we suppose that the initial condition ℎ0satisfies
E[ℎ02119901
] lt infin for some 119901 gt 1 Let 119879 ge 0 be arbitrary Wedefine the Banach space
1198712119901
119879(119867) = 119871
2119901
(Ω times [0 119879] P119879P otimes 119889119905119867) (129)
and prove that the variation of constants equation
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(130)
has a unique solution in the space 1198712119901
119879(119867) This is done in the
following three steps
Step 11 For 119883 isin 1198712119901
119879(119867) we define the process Φ119883 by
(Φ119883)119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(131)
Then the process Φ119883 is well defined Indeed by the growthestimate (9) the linear growth condition (127) and Holderrsquosinequality we have
Consequently there exists an index 119899 isin N such that Φ119899 is
a contraction and hence according to the extension of theBanach fixed point theorem (see Corollary 46) the mappingΦ has a unique fixed point 119883 isin 119871
2119901
119879(119867) This fixed point 119883 is
a solution to the variation of constants equation (130) Since119879 ge 0 was arbitrary there exists a process 119883 which is a solu-tion of the variation of constants equation
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
119905 ge 0
(146)
Step 14 In order to prove that 119883 is a mild solution to (44)it remains to show that 119883 has a continuous version ByLemma 12 the process
119905 997891997888rarr 119878119905ℎ0 119905 ge 0 (147)
is continuous and by Proposition 41 the process
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 119905 ge 0 (148)
is continuous too Moreover for every 119879 ge 0 we have by thelinear growth condition (128) Holderrsquos inequality and since119883 isin 119871
Thus by Proposition 43 the stochastic convolution 120590 ⋆ 119882
given by
(120590 ⋆ 119882)119905= int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0 (150)
has a continuous version and consequently the process119883hasa continuous version too This continuous version is a mildsolution to (44)
Step 2 Now let ℎ0 Ω rarr 119867 be an arbitraryF
0-measurable
random variable We define the sequence (ℎ119899)119899isinN of F
0-
measurable random variables as
ℎ119899
0= ℎ
01ℎ0le119899
119899 isin N (151)
Let 119899 isin N be arbitrary Then as ℎ119899
0is bounded we have
E[ℎ119899
02119901
] lt infin for all 119901 gt 1 By Step 1 the SPDE
119889119883119899
119905= (119860119883
119899
119905+ 120572 (119905 119883
119899
119905)) 119889119905 + 120590 (119905 119883
119899
119905) 119889119882
119905
119883119899
0= ℎ
119899
0
(152)
18 International Journal of Stochastic Analysis
has amild solution119883119899We define the sequence (Ω
119899)119899isinN sub F
0
as
Ω119899=
1003817100381710038171003817ℎ01003817100381710038171003817 le 119899 119899 isin N (153)
Then we have Ω119899sub Ω
119898for 119899 le 119898 Ω = ⋃
119899isinN Ω119899 and
Ω119899sub ℎ
119899
0= ℎ
119898
0 sub ℎ
119899
0= ℎ
0 forall119899 le 119898 (154)
Thus byTheorem 48 we have (up to indistinguishability)
119883119899
1Ω119899
= 119883119898
1Ω119899
forall119899 le 119898 (155)
Consequently the process
119883 = lim119899rarrinfin
119883119899
1Ω119899
(156)
is a well-defined continuous and adapted process and wehave
119883119899
1Ω119899
= 119883119898
1Ω119899
= 1198831Ω119899
forall119899 le 119898 (157)
Furthermore we obtain P-almost surely
119883119905= lim
119899rarrinfin
119883119899
1199051Ω119899
= lim119899rarrinfin
1Ω119899
(119878119905ℎ119899
0+ int
119905
0
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+int
119905
0
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ119899
0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= lim119899rarrinfin
1Ω119899
(119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= 119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(158)
proving that 119883 is a mild solution to (44)
Remark 51 For the proof of Theorem 50 we have usedCorollary 46 which is a slight extension of the Banach fixedpoint theorem Such an idea has been applied for example in[15]
Remark 52 A recent method for proving existence anduniqueness of mild solutions to the SPDE (44) is the methodof the moving frame presented in [6] see also [8] It allowsto reduce SPDE problems to the study of SDEs in infinitedimension In order to apply this method we need that thesemigroup (119878
119905)119905ge0
is a semigroup of pseudocontractions
We close this section with a consequence about theexistence of weak solutions
Corollary 53 Suppose that conditions (125)ndash(128) are ful-filled Let ℎ
0 Ω rarr 119867 be aF
0-measurable random variable
such that E[ℎ02119901
] lt infin for some 119901 gt 1 Then there exists a(up to indistinguishability) unique weak solution 119883 to (44)
Proof According to Proposition 35 every weak solution119883 to(44) is also a mild solution to (44)Therefore the uniquenessof weak solutions to (44) is a consequence of Theorem 48
It remains to prove the existence of a weak solution to(44) Let 119879 ge 0 be arbitrary By Theorem 50 and its proofthere exists a mild solution119883 isin 119871
2119901
119879(119867) to (44) By the linear
growth condition (128) and Holderrsquos inequality we obtain
showing that condition (72) is fulfilled Thus byProposition 37 the process 119883 is also a weak solution to(44)
8 Invariant Manifolds for WeakSolutions to SPDEs
In this section we deal with invariant manifolds for time-homogeneous SPDEs of the type (44) This topic arises fromthe natural desire to express the solutions of the SPDE (44)which generally live in the infinite dimensional Hilbert space119867 by means of a finite dimensional state process and thusto ensure larger analytical tractability Our goal is to findconditions on the generator 119860 and the coefficients 120572 120590 suchthat for every starting point of a finite dimensional submani-fold the solution process stays on the submanifold
We start with the required preliminaries about finitedimensional submanifolds inHilbert spaces In the sequel let119867 be a separable Hilbert space
Definition 54 Let 119898 119896 isin N be positive integers A subsetM sub 119867 is called an 119898-dimensional 119862119896-submanifold of 119867
International Journal of Stochastic Analysis 19
if for every ℎ isin M there exist an open neighborhood 119880 sub 119867
of ℎ an open set 119881 sub R119898 and a mapping 120601 isin 1198622
(119881119867) suchthat
(1) the mapping 120601 119881 rarr 119880 cap M is a homeomorphism(2) for all 119910 isin 119881 the mapping 119863120601(119910) is injective
The mapping 120601 is called a parametrization ofM around ℎ
In what follows let M be an 119898-dimensional 119862119896-
submanifold of 119867
Lemma 55 Let 120601119894
119881119894
rarr 119880119894cap M 119894 = 1 2 be two
parametrizations with119882= 1198801cap119880
2capM = 0Then themapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (160)
is a 119862119896-diffeomorphism
Proof See [16 Lemma 611]
Corollary 56 Let ℎ isin M be arbitrary and let 120601119894
119881119894
rarr
119880119894capM 119894 = 1 2 be two parametrizations ofM around ℎ Then
one has
1198631206011(119910
1) (R
119898
) = 1198631206012(119910
2) (R
119898
) (161)
where 119910119894= 120601
minus1
119894(ℎ) for 119894 = 1 2
Proof Since1198801and119880
2are open neighborhoods of ℎ we have
119882= 1198801cap 119880
2cap M = 0 Thus by Lemma 55 the mapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (162)
is a 119862119896-diffeomorphism Using the chain rule we obtain
1198631206012(119910
2) (R
119898
)
= 119863 (1206011∘ (120601
minus1
1∘ 120601
2)) (119910
2) (R
119898
)
= 1198631206011(119910
1)119863 (120601
minus1
1∘ 120601
2) (119910
2) (R
119898
)
sub 1198631206011(119910
1) (R
119898
)
(163)
and analogously we prove that 1198631206011(119910
1)(R119898
) sub
1198631206012(119910
2)(R119898
)
Definition 57 Let ℎ isin M be arbitraryThe tangent space ofMto ℎ is the subspace
119879ℎM = 119863120601 (119910) (R
119898
) (164)
where 119910 = 120601minus1
(ℎ) and 120601 119881 rarr 119880 cap M denotes aparametrization ofM around ℎ
Remark 58 Note that according to Corollary 56 theDefinition 57 of the tangent space 119879
ℎM does not depend on
the choice of the parametrization 120601 119881 rarr 119880 cap M
Proposition 59 Let ℎ isin M be arbitrary and let 120601 119881 rarr
119880 cap M be a parametrization ofM around ℎ Then there existan open set 119881
0sub 119881 an open neighborhood 119880
0sub 119880 of ℎ and
a mapping 120601 isin 119862119896
119887(R119898
119867) with 120601|1198810
= 120601|1198810
such that 120601|1198810
1198810
rarr 1198800cap M is a parametrization ofM around ℎ too
Proof See [16 Remark 611]
Remark 60 By Proposition 59 we may assume that anyparametrization 120601 119881 rarr 119880 cap M has an extension 120601 isin
119862119896
119887(R119898
119867)
Proposition 61 Let 119863 sub 119867 be a dense subset For every ℎ0isin
M there exist 1205771 120577
119898isin 119863 and a parametrization 120601 119881 rarr
119880 cap M around ℎ0such that
120601 (⟨120577 ℎ⟩) = ℎ forallℎ isin 119880 cap M (165)
where one uses the notation ⟨120577 ℎ⟩ = (⟨1205771 ℎ⟩ ⟨120577
119898 ℎ⟩) isin
R119898
Proof See [16 Proposition 612]
Proposition 62 Let 120601 119881 rarr 119880 cap M be a parametrizationas in Proposition 61 Then the following statements are true
(1) The elements 1205771 120577
119898are linearly independent in 119867
(2) For every ℎ isin 119880 cap M one has the direct sum decom-position
Therefore by Proposition 64 for every 1199100isin R119898 there exists
a (up to indistinguishability) unique weak solution to (192)which according to Proposition 39 is also a strong solutionto (192) The uniqueness of strong solutions to (192) is a con-sequence of Proposition 39 andTheorem 48
Now we are ready to formulate and prove ourmain resultof this section
Theorem 68 The following statements are equivalent
(1) The submanifoldM is locally invariant for (176)(2) One has
M sub D (119860) (195)
120590119895
(ℎ) isin 119879ℎM forallℎ isin M 119886119897119897 119895 isin N (196)
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) isin 119879ℎM forallℎ isin M (197)
(3) The operator 119860 is continuous onM and for each ℎ0isin
M there exists a local strong solution 119883 to (176) withsome lifetime 120591 gt 0 such that
119883119905and120591
isin M forall119905 ge 0 P-119886119897119898119900119904119905 119904119906119903119890119897119910 (198)
Proof (1)rArr(2) Let ℎ isin M be arbitrary By Proposition 61and Remark 60 there exist elements 120577
1 120577
119898isin D(119860
lowast
) anda parametrization 120601 119881 rarr 119880 cap M around ℎ such that theinverse 120601
minus1
119880 cap M rarr 119881 is given by 120601minus1
= ⟨120577 ∙⟩ and 120601
has an extension 120601 isin 1198622
119887(R119898
119867) Since the submanifoldM islocally invariant for (176) there exists a local weak solution119883
to (176) with initial condition ℎ and some lifetime 984858 gt 0 suchthat
119883119905and984858
isin M forall119905 ge 0 P-almost surely (199)
Since119880 is an open neighborhood of ℎ there exists 120598 gt 0 suchthat 119861
120598(ℎ) sub 119880 where 119861
120598(ℎ) denotes the open ball
119861120598(ℎ) = 119892 isin 119867
1003817100381710038171003817119892 minus ℎ1003817100381710038171003817 lt 120598 (200)
We define the stopping time
120591 = 984858 and inf 119905 ge 0 119883119905notin 119861
120598(ℎ) (201)
Since the process119883 has continuous sample paths and satisfies1198830= ℎ we have 120591 gt 0 and P-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (202)
22 International Journal of Stochastic Analysis
Defining the R119898-valued process 119884 = ⟨120577 119883⟩ we have P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (203)
Moreover since 119883 is a weak solution to (176) with initialcondition ℎ setting 119910 = ⟨120577 ℎ⟩ isin 119881 we have P-almost surely
119884119905and120591
= ⟨120577 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120577 120590119895
(119883119904)⟩ 119889120573
119895
119904
= ⟨120577 ℎ⟩ + int
119905and120591
0
120572120601120577
(⟨120577 119883119904⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(⟨120577 119883
119904⟩) 119889120573
119895
119904
= 119910 + int
119905and120591
0
120572120601120577
(119884119904) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(204)
showing that 119884 is a local strong solution to (192) with initialcondition 119910 By Itorsquos formula (Theorem 26) we obtain P-almost surely
119883119905and120591
= 120601 (119884119905and120591
)
= ℎ + int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904)))119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(205)
Now let 120585 isin D(119860lowast
) be arbitrary Then we have P-almostsurely
⟨120585 119883119905and120591
⟩
= ⟨120585 ℎ⟩
+ int
119905and120591
0
⟨120585119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 119863120601 (119884119904) 120590
119895
120601120577(119884
119904)⟩119889120573
119895
119904 119905 ge 0
(206)
On the other hand since 119883 is a local weak solution to (176)with initial condition ℎ and lifetime 120591 we have P-almostsurely for all 119905 ge 0 the identity
⟨120585 119883119905and120591
⟩ = ⟨120585 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904)⟩ 119889120573
119895
119904
(207)
Combining (206) and (207) yields up to indistinguishability
119861 + 119872 = 0 (208)
where the processes 119861 and 119872 are defined as
119861119905= int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩) 119889119904
119905 ge 0
119872119905= sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩ 119889120573
119895
119904
119905 ge 0
(209)
The process 119861 + 119872 is a continuous semimartingale withcanonical decomposition (208) Since the canonical decom-position of a continuous semimartingale is unique up toindistinguishability we deduce that 119861 = 119872 = 0 up toindistinguishability Using the Ito isometry (39) we obtain P-almost surely
int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩
+ ⟨120585 120572 (119883119904) minus 119863120601 (119884
showing that119883 is a local strong solution to (44) with lifetime120591
(3)rArr(1) This implication is a direct consequence ofProposition 29
The results from this section are closely related to the exis-tence of finite dimensional realizations that is the existenceof invariantmanifolds for each starting point ℎ
0 andwe point
out the papers [17ndash22] regarding this topic Furthermore wemention thatTheorem 68 has been extended in [23] to SPDEswith jumps
Acknowledgments
The author is grateful to Daniel Gaigall Georg GrafendorferFlorian Modler and Thomas Salfeld for the valuable com-ments and discussions
References
[1] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions CambridgeUniversity Press CambridgeUK 1992
[2] C Prevot and M Rockner A Concise Course on StochasticPartial Differential Equations Springer Berlin Germany 2007
[3] L Gawarecki and V Mandrekar Stochastic Differential Equa-tions in Infinite Dimensions with Applications to StochasticPartial Differential Equations Springer Berlin Germany 2011
[4] S Peszat and J Zabczyk Stochastic Partial Differential Equationswith Levy Noise Cambridge University Press Cambridge UK2007
[5] S Albeverio V Mandrekar and B Rudiger ldquoExistence of mildsolutions for stochastic differential equations and semilinearequations with non-Gaussian Levy noiserdquo Stochastic Processesand Their Applications vol 119 no 3 pp 835ndash863 2009
[6] D Filipovic S Tappe and J Teichmann ldquoJump-diffusions inHilbert spaces existence stability and numericsrdquo Stochasticsvol 82 no 5 pp 475ndash520 2010
[7] C Marinelli C Prevot and M Rockner ldquoRegular dependenceon initial data for stochastic evolution equations with multi-plicative Poisson noiserdquo Journal of Functional Analysis vol 258no 2 pp 616ndash649 2010
[8] S Tappe ldquoSome refinements of existence results for SPDEsdriven by Wiener processes and Poisson random measuresrdquoInternational Journal of Stochastic Analysis vol 2012 Article ID236327 24 pages 2012
[9] T Nakayama ldquoSupport theorem for mild solutions of SDErsquos inHilbert spacesrdquo Journal of Mathematical Sciences the Universityof Tokyo vol 11 no 3 pp 245ndash311 2004
[10] T Nakayama ldquoViability theorem for SPDErsquos including HJMframeworkrdquo Journal of Mathematical Sciences the University ofTokyo vol 11 no 3 pp 313ndash324 2004
[11] W Rudin Functional Analysis McGraw-Hill New York NYUSA 2nd edition 1991
[12] DWerner Funktionalanalysis Springer Berlin Germany 2007[13] C D Aliprantis and K C Border Infinite Dimensional Analysis
Springer Berlin Germany 3rd edition 2006[14] S N Ethier and T G KurtzMarkov Processes Characterization
andConvergenceWiley Series in Probability and Statistics JohnWiley amp Sons New Jersey NJ USA 1986
[15] V Mandrekar and B Rudiger ldquoExistence and uniqueness ofpath wise solutions for stochastic integral equations driven byLevy noise on separable Banach spacesrdquo Stochastics vol 78 no4 pp 189ndash212 2006
[16] D Filipovic Consistency Problems for Heath-Jarrow-MortonInterest Rate Models Springer Berlin Germany 2001
[17] T Bjork and L Svensson ldquoOn the existence of finite-dimensional realizations for nonlinear forward rate modelsrdquoMathematical Finance vol 11 no 2 pp 205ndash243 2001
[18] T Bjork and C Landen ldquoOn the construction of finite dimen-sional realizations for nonlinear forward rate modelsrdquo Financeand Stochastics vol 6 no 3 pp 303ndash331 2002
[19] D Filipovic and J Teichmann ldquoExistence of invariantmanifoldsfor stochastic equations in infinite dimensionrdquo Journal ofFunctional Analysis vol 197 no 2 pp 398ndash432 2003
[20] D Filipovic and J Teichmann ldquoOn the geometry of the termstructure of interest ratesrdquo Proceedings of The Royal Society ofLondon A vol 460 no 2041 pp 129ndash167 2004
[21] S Tappe ldquoAn alternative approach on the existence of affinerealizations for HJM term structure modelsrdquo Proceedings of
International Journal of Stochastic Analysis 25
TheRoyal Society of LondonA vol 466 no 2122 pp 3033ndash30602010
[22] S Tappe ldquoExistence of affine realizations for Levy term struc-ture modelsrdquo Proceedings of The Royal Society of London A vol468 no 2147 pp 3685ndash3704 2012
[23] D Filipovic S Tappe and J Teichmann ldquoInvariant manifoldswith boundary for jump-diffusionsrdquo httparxivorgabs12021076
Namely the 120582119895are the eigenvalues of 119876 and each 119890
119895is an
eigenvector corresponding to 120582119895 The space H
0= 119876
12
(H)equipped with the inner product
⟨119906 V⟩H0
= ⟨119876minus12
119906 119876minus12V⟩
H (32)
is another separable Hilbert space and (radic120582119895119890119895)119895isinN
is anorthonormal basis According to [1 Proposition 41] thesequence of stochastic processes (120573119895)
119895isinN defined as
120573119895
=1
radic120582119895
⟨119882 119890119895⟩H 119895 isin N (33)
is a sequence of real-valued independent standard Wienerprocesses and one has the expansion
119882 = sum
119895isinN
radic120582119895120573119895
119890119895 (34)
Now let us briefly sketch the construction of the Ito integralwith respect to the Wiener process 119882 Further details can befound in [1 3] We denote by 119871
0
2(119867) = 119871
2(H
0 119867) the space
of Hilbert-Schmidt operators from H0into119867 endowed with
the Hilbert-Schmidt norm
Φ1198710
2(119867)
= (sum
119895isinN
120582119895
10038171003817100381710038171003817Φ119890
119895
10038171003817100381710038171003817
2
)
12
Φ isin 1198710
2(119867) (35)
which itself is a separable Hilbert space The construction ofthe Ito integral is divided into three steps as follows
(1) For every 119871(H 119867)-valued simple process of the form
119883 = 119883010
+
119899
sum
119894=1
1198831198941(119905119894119905119894+1] (36)
with 0 = 1199051
lt sdot sdot sdot lt 119905119899+1
= 119879 and F119905119894
-measurablerandom variables 119883
119894 Ω rarr 119871(H 119867) for 119894 = 1 119899
we set
int
119905
0
119883119904119889119882
119904=
119899
sum
119894=1
119883119894(119882
119905and119905119894+1
minus 119882119905and119905119894
) (37)
(2) For every predictable 1198710
2(119867)-valued process 119883 satis-
fying
E [int
119879
0
1003817100381710038171003817119883119904
1003817100381710038171003817
2
1198710
2(119867)
119889119904] lt infin (38)
we extend the Ito integral int1199050
119883119904119889119882
119904by an extension
argument for linear operators In particular we obtainthe Ito isometry as follows
) is dense in119867 and hencewe obtain P-almost surely
119883119905and120591
= ℎ0+ int
119905and120591
0
(119860119883119904+ 120572 (119904 119883
119904)) 119889119904
+ int
119905and120591
0
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(53)
Consequently the process 119883 is also a local strong solution to(44) with lifetime 120591
Corollary 31 Let M sub D(119860) be a subset such that 119860 iscontinuous on M and let 119883 be a local weak solution to (44)with lifetime 120591 such that
119883119905and120591
isin M forall119905 ge 0 P-119886119897119898119900119904119905 119904119906119903119890119897119910 (54)
Then 119883 is also a local strong solution to (44) with lifetime 120591
Proof Since M sub D(119860) condition (54) implies that (46) isfulfilled Moreover by the continuity of 119860 onM the samplepaths of the process 119860119883 are P-almost surely continuous andhence we obtain (47) Consequently using Proposition 30the process 119883 is also a local strong solution to (44) withlifetime 120591
Proposition 32 Every strong solution119883 to (44) is also a mildsolution to (44)
Proof According to Lemma 8 the domain (D(119860) sdot D(119860))
endowed with the graph norm is a separable Hilbert spacetoo Hence by Lemma 18 for all 119905 ge 0 the function
119891 [0 119905] times D (119860) 997888rarr 119867 119891 (119904 119909) = 119878119905minus119904
119909 (55)
belongs to the class 11986212loc119887
([0 119905] times D(119860)119867) with partialderivatives
Since by Proposition 14 the domain D((119860lowast
)2
) is dense in(D(119860
lowast
) sdot D(119860lowast)) we get P-almost surely for all 120577 isin D(119860
lowast
)
the identity
⟨120577 119883119905⟩ = ⟨120577 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904⟩
(70)
Since by Proposition 14 the domainD(119860lowast
) is dense in119867 weobtain P-almost surely
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 (71)
proving that 119883 is a mild solution to (44)
Remark 36 Now the proof of Proposition 32 is an immediateconsequence of Propositions 29 and 35
We have just seen that every weak solution to (44) is also amild solution Under the following regularity condition (72)the converse of this statement holds true as well
Proposition 37 Let 119883 be a mild solution to (44) such that
120572 (119904 119883119904) minus 120572 (119904 119883
119904)⟩ 119889119904
= ⟨120577 int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904⟩ minus int
119905
0
⟨120577 120572 (119904 119883119904)⟩119889119904
(74)
Due to assumption (72) we may use Fubinirsquos theorem forstochastic integrals (see [3 Theorem 28]) which togetherwith Lemma 18 gives us P-almost surely
Due to assumption (80) we may use Fubinirsquos theorem forstochastic integrals (see [3 Theorem 28]) which togetherwith Lemma 18 gives us P-almost surely
int
119905
0
(119878119905minus119904
120590 (119904 119883119904) minus 120590 (119904 119883
This proves that 119883 is also a strong solution to (44)
The following result shows that for norm continuoussemigroups the concepts of strong weak and mild solutionsare equivalent In particular this applies for finite dimen-sional state spaces
Proposition 39 Suppose that the semigroup (119878119905)119905ge0
is normcontinuous Let 119883 be a stochastic process with 119883
0= ℎ
0 Then
the following statements are equivalent
(1) The process 119883 is a strong solution to (44)
(2) The process 119883 is a weak solution to (44)
(3) The process 119883 is a mild solution to (44)
Proof (1)rArr(2) This implication is a consequence ofProposition 29
(2)rArr(3) This implication is a consequence ofProposition 35
(3)rArr(1) By Proposition 20 we have 119860 isin 119871(119867) and 119878119905=
119890119905119860 119905 ge 0 Furthermore the family (119890
119905119860
)119905isinR is a 119862
0-group on
119867 Therefore and since 119883 is a mild solution to (44) we haveP-almost surely
(2) An element 119909 isin 119864 is called a fixed point of Φ if onehas
Φ (119909) = 119909 (109)
The following result is the well-known Banach fixed pointtheorem Its proof can be found for example in [13Theorem348]
Theorem 45 (The Banach fixed point theorem) Let 119864 be acomplete metric space and let Φ 119864 rarr 119864 be a contractionThen the mapping Φ has a unique fixed point
In this text we will use the following slight extension ofthe Banach fixed point theorem
Corollary 46 Let 119864 be a complete metric space and let Φ
119864 rarr 119864 be a mapping such that for some 119899 isin N the mappingΦ119899 is a contraction Then the mapping Φ has a unique fixed
point
Proof According to the Banach fixed point theorem(Theorem 45) the mapping Φ
119899 has a unique fixed point thatis there exists a unique element 119909 isin 119864 such that Φ119899
(119909) = 119909Therefore we have
Φ (119909) = Φ (Φ119899
(119909)) = Φ119899
(Φ (119909)) (110)
showing thatΦ(119909) is a fixedpoint ofΦ119899 SinceΦ119899 has a uniquefixed point we deduce thatΦ(119909) = 119909 showing that119909 is a fixedpoint of Φ
In order to prove uniqueness let 119910 isin 119864 be another fixedpoint ofΦ that is we haveΦ(119910) = 119910 By induction we obtain
showing that 119910 is a fixed point of Φ119899 Since the mapping Φ119899
has exactly one fixed point we obtain 119909 = 119910
An indispensable tool for proving uniqueness of mildsolutions to (44) will be the following version of Gronwallrsquosinequality see for example [14 Theorem 51]
Lemma 47 (Gronwallrsquos inequality) Let 119879 ge 0 be fixed let119891 [0 119879] rarr R
and hence by the Cauchy-Schwarz inequality the Ito isom-etry (39) the growth estimate (9) from Lemma 10 and thelocal Lipschitz conditions (113) we obtain
Using Gronwallrsquos inequality (see Lemma 47) we deduce that119891 equiv 0 Thus by the continuity of the sample paths of 119883 and119884 we obtain
P(⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ) = 1 forall119899 isin N (120)
and hence by the continuity of the probability measureP weconclude that
P(⋂
119905ge0
119883119905and120591
1Γ= 119884
119905and1205911Γ)
= P(⋂
119899isinN
⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ)
= lim119899rarrinfin
P(⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ) = 1
(121)
which completes the proof
The local Lipschitz conditions (113) are in general notsufficient in order to ensure the existence of mild solutions tothe SPDE (44) Now we will prove that the existence of mildsolutions follows from global Lipschitz and linear growthconditions on 120572 and 120590 For this we recall an auxiliary resultwhich extends the Ito isometry (39)
Lemma 49 Let 119879 ge 0 be arbitrary and let 119883 = (119883119905)119905isin[0119879]
for all 119905 ge 0 and all ℎ isin 119867 Then for every F0-measurable
random variable ℎ0
Ω rarr 119867 there exists a (up toindistinguishability) unique mild solution 119883 to (44)
Proof The uniqueness of mild solutions to (44) is a directconsequence of Theorem 48 and hence we may concentrateon the existence proof which we divide into the followingseveral steps
Step 1 First we suppose that the initial condition ℎ0satisfies
E[ℎ02119901
] lt infin for some 119901 gt 1 Let 119879 ge 0 be arbitrary Wedefine the Banach space
1198712119901
119879(119867) = 119871
2119901
(Ω times [0 119879] P119879P otimes 119889119905119867) (129)
and prove that the variation of constants equation
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(130)
has a unique solution in the space 1198712119901
119879(119867) This is done in the
following three steps
Step 11 For 119883 isin 1198712119901
119879(119867) we define the process Φ119883 by
(Φ119883)119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(131)
Then the process Φ119883 is well defined Indeed by the growthestimate (9) the linear growth condition (127) and Holderrsquosinequality we have
Consequently there exists an index 119899 isin N such that Φ119899 is
a contraction and hence according to the extension of theBanach fixed point theorem (see Corollary 46) the mappingΦ has a unique fixed point 119883 isin 119871
2119901
119879(119867) This fixed point 119883 is
a solution to the variation of constants equation (130) Since119879 ge 0 was arbitrary there exists a process 119883 which is a solu-tion of the variation of constants equation
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
119905 ge 0
(146)
Step 14 In order to prove that 119883 is a mild solution to (44)it remains to show that 119883 has a continuous version ByLemma 12 the process
119905 997891997888rarr 119878119905ℎ0 119905 ge 0 (147)
is continuous and by Proposition 41 the process
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 119905 ge 0 (148)
is continuous too Moreover for every 119879 ge 0 we have by thelinear growth condition (128) Holderrsquos inequality and since119883 isin 119871
Thus by Proposition 43 the stochastic convolution 120590 ⋆ 119882
given by
(120590 ⋆ 119882)119905= int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0 (150)
has a continuous version and consequently the process119883hasa continuous version too This continuous version is a mildsolution to (44)
Step 2 Now let ℎ0 Ω rarr 119867 be an arbitraryF
0-measurable
random variable We define the sequence (ℎ119899)119899isinN of F
0-
measurable random variables as
ℎ119899
0= ℎ
01ℎ0le119899
119899 isin N (151)
Let 119899 isin N be arbitrary Then as ℎ119899
0is bounded we have
E[ℎ119899
02119901
] lt infin for all 119901 gt 1 By Step 1 the SPDE
119889119883119899
119905= (119860119883
119899
119905+ 120572 (119905 119883
119899
119905)) 119889119905 + 120590 (119905 119883
119899
119905) 119889119882
119905
119883119899
0= ℎ
119899
0
(152)
18 International Journal of Stochastic Analysis
has amild solution119883119899We define the sequence (Ω
119899)119899isinN sub F
0
as
Ω119899=
1003817100381710038171003817ℎ01003817100381710038171003817 le 119899 119899 isin N (153)
Then we have Ω119899sub Ω
119898for 119899 le 119898 Ω = ⋃
119899isinN Ω119899 and
Ω119899sub ℎ
119899
0= ℎ
119898
0 sub ℎ
119899
0= ℎ
0 forall119899 le 119898 (154)
Thus byTheorem 48 we have (up to indistinguishability)
119883119899
1Ω119899
= 119883119898
1Ω119899
forall119899 le 119898 (155)
Consequently the process
119883 = lim119899rarrinfin
119883119899
1Ω119899
(156)
is a well-defined continuous and adapted process and wehave
119883119899
1Ω119899
= 119883119898
1Ω119899
= 1198831Ω119899
forall119899 le 119898 (157)
Furthermore we obtain P-almost surely
119883119905= lim
119899rarrinfin
119883119899
1199051Ω119899
= lim119899rarrinfin
1Ω119899
(119878119905ℎ119899
0+ int
119905
0
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+int
119905
0
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ119899
0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= lim119899rarrinfin
1Ω119899
(119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= 119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(158)
proving that 119883 is a mild solution to (44)
Remark 51 For the proof of Theorem 50 we have usedCorollary 46 which is a slight extension of the Banach fixedpoint theorem Such an idea has been applied for example in[15]
Remark 52 A recent method for proving existence anduniqueness of mild solutions to the SPDE (44) is the methodof the moving frame presented in [6] see also [8] It allowsto reduce SPDE problems to the study of SDEs in infinitedimension In order to apply this method we need that thesemigroup (119878
119905)119905ge0
is a semigroup of pseudocontractions
We close this section with a consequence about theexistence of weak solutions
Corollary 53 Suppose that conditions (125)ndash(128) are ful-filled Let ℎ
0 Ω rarr 119867 be aF
0-measurable random variable
such that E[ℎ02119901
] lt infin for some 119901 gt 1 Then there exists a(up to indistinguishability) unique weak solution 119883 to (44)
Proof According to Proposition 35 every weak solution119883 to(44) is also a mild solution to (44)Therefore the uniquenessof weak solutions to (44) is a consequence of Theorem 48
It remains to prove the existence of a weak solution to(44) Let 119879 ge 0 be arbitrary By Theorem 50 and its proofthere exists a mild solution119883 isin 119871
2119901
119879(119867) to (44) By the linear
growth condition (128) and Holderrsquos inequality we obtain
showing that condition (72) is fulfilled Thus byProposition 37 the process 119883 is also a weak solution to(44)
8 Invariant Manifolds for WeakSolutions to SPDEs
In this section we deal with invariant manifolds for time-homogeneous SPDEs of the type (44) This topic arises fromthe natural desire to express the solutions of the SPDE (44)which generally live in the infinite dimensional Hilbert space119867 by means of a finite dimensional state process and thusto ensure larger analytical tractability Our goal is to findconditions on the generator 119860 and the coefficients 120572 120590 suchthat for every starting point of a finite dimensional submani-fold the solution process stays on the submanifold
We start with the required preliminaries about finitedimensional submanifolds inHilbert spaces In the sequel let119867 be a separable Hilbert space
Definition 54 Let 119898 119896 isin N be positive integers A subsetM sub 119867 is called an 119898-dimensional 119862119896-submanifold of 119867
International Journal of Stochastic Analysis 19
if for every ℎ isin M there exist an open neighborhood 119880 sub 119867
of ℎ an open set 119881 sub R119898 and a mapping 120601 isin 1198622
(119881119867) suchthat
(1) the mapping 120601 119881 rarr 119880 cap M is a homeomorphism(2) for all 119910 isin 119881 the mapping 119863120601(119910) is injective
The mapping 120601 is called a parametrization ofM around ℎ
In what follows let M be an 119898-dimensional 119862119896-
submanifold of 119867
Lemma 55 Let 120601119894
119881119894
rarr 119880119894cap M 119894 = 1 2 be two
parametrizations with119882= 1198801cap119880
2capM = 0Then themapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (160)
is a 119862119896-diffeomorphism
Proof See [16 Lemma 611]
Corollary 56 Let ℎ isin M be arbitrary and let 120601119894
119881119894
rarr
119880119894capM 119894 = 1 2 be two parametrizations ofM around ℎ Then
one has
1198631206011(119910
1) (R
119898
) = 1198631206012(119910
2) (R
119898
) (161)
where 119910119894= 120601
minus1
119894(ℎ) for 119894 = 1 2
Proof Since1198801and119880
2are open neighborhoods of ℎ we have
119882= 1198801cap 119880
2cap M = 0 Thus by Lemma 55 the mapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (162)
is a 119862119896-diffeomorphism Using the chain rule we obtain
1198631206012(119910
2) (R
119898
)
= 119863 (1206011∘ (120601
minus1
1∘ 120601
2)) (119910
2) (R
119898
)
= 1198631206011(119910
1)119863 (120601
minus1
1∘ 120601
2) (119910
2) (R
119898
)
sub 1198631206011(119910
1) (R
119898
)
(163)
and analogously we prove that 1198631206011(119910
1)(R119898
) sub
1198631206012(119910
2)(R119898
)
Definition 57 Let ℎ isin M be arbitraryThe tangent space ofMto ℎ is the subspace
119879ℎM = 119863120601 (119910) (R
119898
) (164)
where 119910 = 120601minus1
(ℎ) and 120601 119881 rarr 119880 cap M denotes aparametrization ofM around ℎ
Remark 58 Note that according to Corollary 56 theDefinition 57 of the tangent space 119879
ℎM does not depend on
the choice of the parametrization 120601 119881 rarr 119880 cap M
Proposition 59 Let ℎ isin M be arbitrary and let 120601 119881 rarr
119880 cap M be a parametrization ofM around ℎ Then there existan open set 119881
0sub 119881 an open neighborhood 119880
0sub 119880 of ℎ and
a mapping 120601 isin 119862119896
119887(R119898
119867) with 120601|1198810
= 120601|1198810
such that 120601|1198810
1198810
rarr 1198800cap M is a parametrization ofM around ℎ too
Proof See [16 Remark 611]
Remark 60 By Proposition 59 we may assume that anyparametrization 120601 119881 rarr 119880 cap M has an extension 120601 isin
119862119896
119887(R119898
119867)
Proposition 61 Let 119863 sub 119867 be a dense subset For every ℎ0isin
M there exist 1205771 120577
119898isin 119863 and a parametrization 120601 119881 rarr
119880 cap M around ℎ0such that
120601 (⟨120577 ℎ⟩) = ℎ forallℎ isin 119880 cap M (165)
where one uses the notation ⟨120577 ℎ⟩ = (⟨1205771 ℎ⟩ ⟨120577
119898 ℎ⟩) isin
R119898
Proof See [16 Proposition 612]
Proposition 62 Let 120601 119881 rarr 119880 cap M be a parametrizationas in Proposition 61 Then the following statements are true
(1) The elements 1205771 120577
119898are linearly independent in 119867
(2) For every ℎ isin 119880 cap M one has the direct sum decom-position
Therefore by Proposition 64 for every 1199100isin R119898 there exists
a (up to indistinguishability) unique weak solution to (192)which according to Proposition 39 is also a strong solutionto (192) The uniqueness of strong solutions to (192) is a con-sequence of Proposition 39 andTheorem 48
Now we are ready to formulate and prove ourmain resultof this section
Theorem 68 The following statements are equivalent
(1) The submanifoldM is locally invariant for (176)(2) One has
M sub D (119860) (195)
120590119895
(ℎ) isin 119879ℎM forallℎ isin M 119886119897119897 119895 isin N (196)
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) isin 119879ℎM forallℎ isin M (197)
(3) The operator 119860 is continuous onM and for each ℎ0isin
M there exists a local strong solution 119883 to (176) withsome lifetime 120591 gt 0 such that
119883119905and120591
isin M forall119905 ge 0 P-119886119897119898119900119904119905 119904119906119903119890119897119910 (198)
Proof (1)rArr(2) Let ℎ isin M be arbitrary By Proposition 61and Remark 60 there exist elements 120577
1 120577
119898isin D(119860
lowast
) anda parametrization 120601 119881 rarr 119880 cap M around ℎ such that theinverse 120601
minus1
119880 cap M rarr 119881 is given by 120601minus1
= ⟨120577 ∙⟩ and 120601
has an extension 120601 isin 1198622
119887(R119898
119867) Since the submanifoldM islocally invariant for (176) there exists a local weak solution119883
to (176) with initial condition ℎ and some lifetime 984858 gt 0 suchthat
119883119905and984858
isin M forall119905 ge 0 P-almost surely (199)
Since119880 is an open neighborhood of ℎ there exists 120598 gt 0 suchthat 119861
120598(ℎ) sub 119880 where 119861
120598(ℎ) denotes the open ball
119861120598(ℎ) = 119892 isin 119867
1003817100381710038171003817119892 minus ℎ1003817100381710038171003817 lt 120598 (200)
We define the stopping time
120591 = 984858 and inf 119905 ge 0 119883119905notin 119861
120598(ℎ) (201)
Since the process119883 has continuous sample paths and satisfies1198830= ℎ we have 120591 gt 0 and P-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (202)
22 International Journal of Stochastic Analysis
Defining the R119898-valued process 119884 = ⟨120577 119883⟩ we have P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (203)
Moreover since 119883 is a weak solution to (176) with initialcondition ℎ setting 119910 = ⟨120577 ℎ⟩ isin 119881 we have P-almost surely
119884119905and120591
= ⟨120577 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120577 120590119895
(119883119904)⟩ 119889120573
119895
119904
= ⟨120577 ℎ⟩ + int
119905and120591
0
120572120601120577
(⟨120577 119883119904⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(⟨120577 119883
119904⟩) 119889120573
119895
119904
= 119910 + int
119905and120591
0
120572120601120577
(119884119904) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(204)
showing that 119884 is a local strong solution to (192) with initialcondition 119910 By Itorsquos formula (Theorem 26) we obtain P-almost surely
119883119905and120591
= 120601 (119884119905and120591
)
= ℎ + int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904)))119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(205)
Now let 120585 isin D(119860lowast
) be arbitrary Then we have P-almostsurely
⟨120585 119883119905and120591
⟩
= ⟨120585 ℎ⟩
+ int
119905and120591
0
⟨120585119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 119863120601 (119884119904) 120590
119895
120601120577(119884
119904)⟩119889120573
119895
119904 119905 ge 0
(206)
On the other hand since 119883 is a local weak solution to (176)with initial condition ℎ and lifetime 120591 we have P-almostsurely for all 119905 ge 0 the identity
⟨120585 119883119905and120591
⟩ = ⟨120585 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904)⟩ 119889120573
119895
119904
(207)
Combining (206) and (207) yields up to indistinguishability
119861 + 119872 = 0 (208)
where the processes 119861 and 119872 are defined as
119861119905= int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩) 119889119904
119905 ge 0
119872119905= sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩ 119889120573
119895
119904
119905 ge 0
(209)
The process 119861 + 119872 is a continuous semimartingale withcanonical decomposition (208) Since the canonical decom-position of a continuous semimartingale is unique up toindistinguishability we deduce that 119861 = 119872 = 0 up toindistinguishability Using the Ito isometry (39) we obtain P-almost surely
int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩
+ ⟨120585 120572 (119883119904) minus 119863120601 (119884
showing that119883 is a local strong solution to (44) with lifetime120591
(3)rArr(1) This implication is a direct consequence ofProposition 29
The results from this section are closely related to the exis-tence of finite dimensional realizations that is the existenceof invariantmanifolds for each starting point ℎ
0 andwe point
out the papers [17ndash22] regarding this topic Furthermore wemention thatTheorem 68 has been extended in [23] to SPDEswith jumps
Acknowledgments
The author is grateful to Daniel Gaigall Georg GrafendorferFlorian Modler and Thomas Salfeld for the valuable com-ments and discussions
References
[1] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions CambridgeUniversity Press CambridgeUK 1992
[2] C Prevot and M Rockner A Concise Course on StochasticPartial Differential Equations Springer Berlin Germany 2007
[3] L Gawarecki and V Mandrekar Stochastic Differential Equa-tions in Infinite Dimensions with Applications to StochasticPartial Differential Equations Springer Berlin Germany 2011
[4] S Peszat and J Zabczyk Stochastic Partial Differential Equationswith Levy Noise Cambridge University Press Cambridge UK2007
[5] S Albeverio V Mandrekar and B Rudiger ldquoExistence of mildsolutions for stochastic differential equations and semilinearequations with non-Gaussian Levy noiserdquo Stochastic Processesand Their Applications vol 119 no 3 pp 835ndash863 2009
[6] D Filipovic S Tappe and J Teichmann ldquoJump-diffusions inHilbert spaces existence stability and numericsrdquo Stochasticsvol 82 no 5 pp 475ndash520 2010
[7] C Marinelli C Prevot and M Rockner ldquoRegular dependenceon initial data for stochastic evolution equations with multi-plicative Poisson noiserdquo Journal of Functional Analysis vol 258no 2 pp 616ndash649 2010
[8] S Tappe ldquoSome refinements of existence results for SPDEsdriven by Wiener processes and Poisson random measuresrdquoInternational Journal of Stochastic Analysis vol 2012 Article ID236327 24 pages 2012
[9] T Nakayama ldquoSupport theorem for mild solutions of SDErsquos inHilbert spacesrdquo Journal of Mathematical Sciences the Universityof Tokyo vol 11 no 3 pp 245ndash311 2004
[10] T Nakayama ldquoViability theorem for SPDErsquos including HJMframeworkrdquo Journal of Mathematical Sciences the University ofTokyo vol 11 no 3 pp 313ndash324 2004
[11] W Rudin Functional Analysis McGraw-Hill New York NYUSA 2nd edition 1991
[12] DWerner Funktionalanalysis Springer Berlin Germany 2007[13] C D Aliprantis and K C Border Infinite Dimensional Analysis
Springer Berlin Germany 3rd edition 2006[14] S N Ethier and T G KurtzMarkov Processes Characterization
andConvergenceWiley Series in Probability and Statistics JohnWiley amp Sons New Jersey NJ USA 1986
[15] V Mandrekar and B Rudiger ldquoExistence and uniqueness ofpath wise solutions for stochastic integral equations driven byLevy noise on separable Banach spacesrdquo Stochastics vol 78 no4 pp 189ndash212 2006
[16] D Filipovic Consistency Problems for Heath-Jarrow-MortonInterest Rate Models Springer Berlin Germany 2001
[17] T Bjork and L Svensson ldquoOn the existence of finite-dimensional realizations for nonlinear forward rate modelsrdquoMathematical Finance vol 11 no 2 pp 205ndash243 2001
[18] T Bjork and C Landen ldquoOn the construction of finite dimen-sional realizations for nonlinear forward rate modelsrdquo Financeand Stochastics vol 6 no 3 pp 303ndash331 2002
[19] D Filipovic and J Teichmann ldquoExistence of invariantmanifoldsfor stochastic equations in infinite dimensionrdquo Journal ofFunctional Analysis vol 197 no 2 pp 398ndash432 2003
[20] D Filipovic and J Teichmann ldquoOn the geometry of the termstructure of interest ratesrdquo Proceedings of The Royal Society ofLondon A vol 460 no 2041 pp 129ndash167 2004
[21] S Tappe ldquoAn alternative approach on the existence of affinerealizations for HJM term structure modelsrdquo Proceedings of
International Journal of Stochastic Analysis 25
TheRoyal Society of LondonA vol 466 no 2122 pp 3033ndash30602010
[22] S Tappe ldquoExistence of affine realizations for Levy term struc-ture modelsrdquo Proceedings of The Royal Society of London A vol468 no 2147 pp 3685ndash3704 2012
[23] D Filipovic S Tappe and J Teichmann ldquoInvariant manifoldswith boundary for jump-diffusionsrdquo httparxivorgabs12021076
) is dense in119867 and hencewe obtain P-almost surely
119883119905and120591
= ℎ0+ int
119905and120591
0
(119860119883119904+ 120572 (119904 119883
119904)) 119889119904
+ int
119905and120591
0
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(53)
Consequently the process 119883 is also a local strong solution to(44) with lifetime 120591
Corollary 31 Let M sub D(119860) be a subset such that 119860 iscontinuous on M and let 119883 be a local weak solution to (44)with lifetime 120591 such that
119883119905and120591
isin M forall119905 ge 0 P-119886119897119898119900119904119905 119904119906119903119890119897119910 (54)
Then 119883 is also a local strong solution to (44) with lifetime 120591
Proof Since M sub D(119860) condition (54) implies that (46) isfulfilled Moreover by the continuity of 119860 onM the samplepaths of the process 119860119883 are P-almost surely continuous andhence we obtain (47) Consequently using Proposition 30the process 119883 is also a local strong solution to (44) withlifetime 120591
Proposition 32 Every strong solution119883 to (44) is also a mildsolution to (44)
Proof According to Lemma 8 the domain (D(119860) sdot D(119860))
endowed with the graph norm is a separable Hilbert spacetoo Hence by Lemma 18 for all 119905 ge 0 the function
119891 [0 119905] times D (119860) 997888rarr 119867 119891 (119904 119909) = 119878119905minus119904
119909 (55)
belongs to the class 11986212loc119887
([0 119905] times D(119860)119867) with partialderivatives
Since by Proposition 14 the domain D((119860lowast
)2
) is dense in(D(119860
lowast
) sdot D(119860lowast)) we get P-almost surely for all 120577 isin D(119860
lowast
)
the identity
⟨120577 119883119905⟩ = ⟨120577 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904⟩
(70)
Since by Proposition 14 the domainD(119860lowast
) is dense in119867 weobtain P-almost surely
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 (71)
proving that 119883 is a mild solution to (44)
Remark 36 Now the proof of Proposition 32 is an immediateconsequence of Propositions 29 and 35
We have just seen that every weak solution to (44) is also amild solution Under the following regularity condition (72)the converse of this statement holds true as well
Proposition 37 Let 119883 be a mild solution to (44) such that
120572 (119904 119883119904) minus 120572 (119904 119883
119904)⟩ 119889119904
= ⟨120577 int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904⟩ minus int
119905
0
⟨120577 120572 (119904 119883119904)⟩119889119904
(74)
Due to assumption (72) we may use Fubinirsquos theorem forstochastic integrals (see [3 Theorem 28]) which togetherwith Lemma 18 gives us P-almost surely
Due to assumption (80) we may use Fubinirsquos theorem forstochastic integrals (see [3 Theorem 28]) which togetherwith Lemma 18 gives us P-almost surely
int
119905
0
(119878119905minus119904
120590 (119904 119883119904) minus 120590 (119904 119883
This proves that 119883 is also a strong solution to (44)
The following result shows that for norm continuoussemigroups the concepts of strong weak and mild solutionsare equivalent In particular this applies for finite dimen-sional state spaces
Proposition 39 Suppose that the semigroup (119878119905)119905ge0
is normcontinuous Let 119883 be a stochastic process with 119883
0= ℎ
0 Then
the following statements are equivalent
(1) The process 119883 is a strong solution to (44)
(2) The process 119883 is a weak solution to (44)
(3) The process 119883 is a mild solution to (44)
Proof (1)rArr(2) This implication is a consequence ofProposition 29
(2)rArr(3) This implication is a consequence ofProposition 35
(3)rArr(1) By Proposition 20 we have 119860 isin 119871(119867) and 119878119905=
119890119905119860 119905 ge 0 Furthermore the family (119890
119905119860
)119905isinR is a 119862
0-group on
119867 Therefore and since 119883 is a mild solution to (44) we haveP-almost surely
(2) An element 119909 isin 119864 is called a fixed point of Φ if onehas
Φ (119909) = 119909 (109)
The following result is the well-known Banach fixed pointtheorem Its proof can be found for example in [13Theorem348]
Theorem 45 (The Banach fixed point theorem) Let 119864 be acomplete metric space and let Φ 119864 rarr 119864 be a contractionThen the mapping Φ has a unique fixed point
In this text we will use the following slight extension ofthe Banach fixed point theorem
Corollary 46 Let 119864 be a complete metric space and let Φ
119864 rarr 119864 be a mapping such that for some 119899 isin N the mappingΦ119899 is a contraction Then the mapping Φ has a unique fixed
point
Proof According to the Banach fixed point theorem(Theorem 45) the mapping Φ
119899 has a unique fixed point thatis there exists a unique element 119909 isin 119864 such that Φ119899
(119909) = 119909Therefore we have
Φ (119909) = Φ (Φ119899
(119909)) = Φ119899
(Φ (119909)) (110)
showing thatΦ(119909) is a fixedpoint ofΦ119899 SinceΦ119899 has a uniquefixed point we deduce thatΦ(119909) = 119909 showing that119909 is a fixedpoint of Φ
In order to prove uniqueness let 119910 isin 119864 be another fixedpoint ofΦ that is we haveΦ(119910) = 119910 By induction we obtain
showing that 119910 is a fixed point of Φ119899 Since the mapping Φ119899
has exactly one fixed point we obtain 119909 = 119910
An indispensable tool for proving uniqueness of mildsolutions to (44) will be the following version of Gronwallrsquosinequality see for example [14 Theorem 51]
Lemma 47 (Gronwallrsquos inequality) Let 119879 ge 0 be fixed let119891 [0 119879] rarr R
and hence by the Cauchy-Schwarz inequality the Ito isom-etry (39) the growth estimate (9) from Lemma 10 and thelocal Lipschitz conditions (113) we obtain
Using Gronwallrsquos inequality (see Lemma 47) we deduce that119891 equiv 0 Thus by the continuity of the sample paths of 119883 and119884 we obtain
P(⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ) = 1 forall119899 isin N (120)
and hence by the continuity of the probability measureP weconclude that
P(⋂
119905ge0
119883119905and120591
1Γ= 119884
119905and1205911Γ)
= P(⋂
119899isinN
⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ)
= lim119899rarrinfin
P(⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ) = 1
(121)
which completes the proof
The local Lipschitz conditions (113) are in general notsufficient in order to ensure the existence of mild solutions tothe SPDE (44) Now we will prove that the existence of mildsolutions follows from global Lipschitz and linear growthconditions on 120572 and 120590 For this we recall an auxiliary resultwhich extends the Ito isometry (39)
Lemma 49 Let 119879 ge 0 be arbitrary and let 119883 = (119883119905)119905isin[0119879]
for all 119905 ge 0 and all ℎ isin 119867 Then for every F0-measurable
random variable ℎ0
Ω rarr 119867 there exists a (up toindistinguishability) unique mild solution 119883 to (44)
Proof The uniqueness of mild solutions to (44) is a directconsequence of Theorem 48 and hence we may concentrateon the existence proof which we divide into the followingseveral steps
Step 1 First we suppose that the initial condition ℎ0satisfies
E[ℎ02119901
] lt infin for some 119901 gt 1 Let 119879 ge 0 be arbitrary Wedefine the Banach space
1198712119901
119879(119867) = 119871
2119901
(Ω times [0 119879] P119879P otimes 119889119905119867) (129)
and prove that the variation of constants equation
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(130)
has a unique solution in the space 1198712119901
119879(119867) This is done in the
following three steps
Step 11 For 119883 isin 1198712119901
119879(119867) we define the process Φ119883 by
(Φ119883)119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(131)
Then the process Φ119883 is well defined Indeed by the growthestimate (9) the linear growth condition (127) and Holderrsquosinequality we have
Consequently there exists an index 119899 isin N such that Φ119899 is
a contraction and hence according to the extension of theBanach fixed point theorem (see Corollary 46) the mappingΦ has a unique fixed point 119883 isin 119871
2119901
119879(119867) This fixed point 119883 is
a solution to the variation of constants equation (130) Since119879 ge 0 was arbitrary there exists a process 119883 which is a solu-tion of the variation of constants equation
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
119905 ge 0
(146)
Step 14 In order to prove that 119883 is a mild solution to (44)it remains to show that 119883 has a continuous version ByLemma 12 the process
119905 997891997888rarr 119878119905ℎ0 119905 ge 0 (147)
is continuous and by Proposition 41 the process
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 119905 ge 0 (148)
is continuous too Moreover for every 119879 ge 0 we have by thelinear growth condition (128) Holderrsquos inequality and since119883 isin 119871
Thus by Proposition 43 the stochastic convolution 120590 ⋆ 119882
given by
(120590 ⋆ 119882)119905= int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0 (150)
has a continuous version and consequently the process119883hasa continuous version too This continuous version is a mildsolution to (44)
Step 2 Now let ℎ0 Ω rarr 119867 be an arbitraryF
0-measurable
random variable We define the sequence (ℎ119899)119899isinN of F
0-
measurable random variables as
ℎ119899
0= ℎ
01ℎ0le119899
119899 isin N (151)
Let 119899 isin N be arbitrary Then as ℎ119899
0is bounded we have
E[ℎ119899
02119901
] lt infin for all 119901 gt 1 By Step 1 the SPDE
119889119883119899
119905= (119860119883
119899
119905+ 120572 (119905 119883
119899
119905)) 119889119905 + 120590 (119905 119883
119899
119905) 119889119882
119905
119883119899
0= ℎ
119899
0
(152)
18 International Journal of Stochastic Analysis
has amild solution119883119899We define the sequence (Ω
119899)119899isinN sub F
0
as
Ω119899=
1003817100381710038171003817ℎ01003817100381710038171003817 le 119899 119899 isin N (153)
Then we have Ω119899sub Ω
119898for 119899 le 119898 Ω = ⋃
119899isinN Ω119899 and
Ω119899sub ℎ
119899
0= ℎ
119898
0 sub ℎ
119899
0= ℎ
0 forall119899 le 119898 (154)
Thus byTheorem 48 we have (up to indistinguishability)
119883119899
1Ω119899
= 119883119898
1Ω119899
forall119899 le 119898 (155)
Consequently the process
119883 = lim119899rarrinfin
119883119899
1Ω119899
(156)
is a well-defined continuous and adapted process and wehave
119883119899
1Ω119899
= 119883119898
1Ω119899
= 1198831Ω119899
forall119899 le 119898 (157)
Furthermore we obtain P-almost surely
119883119905= lim
119899rarrinfin
119883119899
1199051Ω119899
= lim119899rarrinfin
1Ω119899
(119878119905ℎ119899
0+ int
119905
0
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+int
119905
0
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ119899
0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= lim119899rarrinfin
1Ω119899
(119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= 119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(158)
proving that 119883 is a mild solution to (44)
Remark 51 For the proof of Theorem 50 we have usedCorollary 46 which is a slight extension of the Banach fixedpoint theorem Such an idea has been applied for example in[15]
Remark 52 A recent method for proving existence anduniqueness of mild solutions to the SPDE (44) is the methodof the moving frame presented in [6] see also [8] It allowsto reduce SPDE problems to the study of SDEs in infinitedimension In order to apply this method we need that thesemigroup (119878
119905)119905ge0
is a semigroup of pseudocontractions
We close this section with a consequence about theexistence of weak solutions
Corollary 53 Suppose that conditions (125)ndash(128) are ful-filled Let ℎ
0 Ω rarr 119867 be aF
0-measurable random variable
such that E[ℎ02119901
] lt infin for some 119901 gt 1 Then there exists a(up to indistinguishability) unique weak solution 119883 to (44)
Proof According to Proposition 35 every weak solution119883 to(44) is also a mild solution to (44)Therefore the uniquenessof weak solutions to (44) is a consequence of Theorem 48
It remains to prove the existence of a weak solution to(44) Let 119879 ge 0 be arbitrary By Theorem 50 and its proofthere exists a mild solution119883 isin 119871
2119901
119879(119867) to (44) By the linear
growth condition (128) and Holderrsquos inequality we obtain
showing that condition (72) is fulfilled Thus byProposition 37 the process 119883 is also a weak solution to(44)
8 Invariant Manifolds for WeakSolutions to SPDEs
In this section we deal with invariant manifolds for time-homogeneous SPDEs of the type (44) This topic arises fromthe natural desire to express the solutions of the SPDE (44)which generally live in the infinite dimensional Hilbert space119867 by means of a finite dimensional state process and thusto ensure larger analytical tractability Our goal is to findconditions on the generator 119860 and the coefficients 120572 120590 suchthat for every starting point of a finite dimensional submani-fold the solution process stays on the submanifold
We start with the required preliminaries about finitedimensional submanifolds inHilbert spaces In the sequel let119867 be a separable Hilbert space
Definition 54 Let 119898 119896 isin N be positive integers A subsetM sub 119867 is called an 119898-dimensional 119862119896-submanifold of 119867
International Journal of Stochastic Analysis 19
if for every ℎ isin M there exist an open neighborhood 119880 sub 119867
of ℎ an open set 119881 sub R119898 and a mapping 120601 isin 1198622
(119881119867) suchthat
(1) the mapping 120601 119881 rarr 119880 cap M is a homeomorphism(2) for all 119910 isin 119881 the mapping 119863120601(119910) is injective
The mapping 120601 is called a parametrization ofM around ℎ
In what follows let M be an 119898-dimensional 119862119896-
submanifold of 119867
Lemma 55 Let 120601119894
119881119894
rarr 119880119894cap M 119894 = 1 2 be two
parametrizations with119882= 1198801cap119880
2capM = 0Then themapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (160)
is a 119862119896-diffeomorphism
Proof See [16 Lemma 611]
Corollary 56 Let ℎ isin M be arbitrary and let 120601119894
119881119894
rarr
119880119894capM 119894 = 1 2 be two parametrizations ofM around ℎ Then
one has
1198631206011(119910
1) (R
119898
) = 1198631206012(119910
2) (R
119898
) (161)
where 119910119894= 120601
minus1
119894(ℎ) for 119894 = 1 2
Proof Since1198801and119880
2are open neighborhoods of ℎ we have
119882= 1198801cap 119880
2cap M = 0 Thus by Lemma 55 the mapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (162)
is a 119862119896-diffeomorphism Using the chain rule we obtain
1198631206012(119910
2) (R
119898
)
= 119863 (1206011∘ (120601
minus1
1∘ 120601
2)) (119910
2) (R
119898
)
= 1198631206011(119910
1)119863 (120601
minus1
1∘ 120601
2) (119910
2) (R
119898
)
sub 1198631206011(119910
1) (R
119898
)
(163)
and analogously we prove that 1198631206011(119910
1)(R119898
) sub
1198631206012(119910
2)(R119898
)
Definition 57 Let ℎ isin M be arbitraryThe tangent space ofMto ℎ is the subspace
119879ℎM = 119863120601 (119910) (R
119898
) (164)
where 119910 = 120601minus1
(ℎ) and 120601 119881 rarr 119880 cap M denotes aparametrization ofM around ℎ
Remark 58 Note that according to Corollary 56 theDefinition 57 of the tangent space 119879
ℎM does not depend on
the choice of the parametrization 120601 119881 rarr 119880 cap M
Proposition 59 Let ℎ isin M be arbitrary and let 120601 119881 rarr
119880 cap M be a parametrization ofM around ℎ Then there existan open set 119881
0sub 119881 an open neighborhood 119880
0sub 119880 of ℎ and
a mapping 120601 isin 119862119896
119887(R119898
119867) with 120601|1198810
= 120601|1198810
such that 120601|1198810
1198810
rarr 1198800cap M is a parametrization ofM around ℎ too
Proof See [16 Remark 611]
Remark 60 By Proposition 59 we may assume that anyparametrization 120601 119881 rarr 119880 cap M has an extension 120601 isin
119862119896
119887(R119898
119867)
Proposition 61 Let 119863 sub 119867 be a dense subset For every ℎ0isin
M there exist 1205771 120577
119898isin 119863 and a parametrization 120601 119881 rarr
119880 cap M around ℎ0such that
120601 (⟨120577 ℎ⟩) = ℎ forallℎ isin 119880 cap M (165)
where one uses the notation ⟨120577 ℎ⟩ = (⟨1205771 ℎ⟩ ⟨120577
119898 ℎ⟩) isin
R119898
Proof See [16 Proposition 612]
Proposition 62 Let 120601 119881 rarr 119880 cap M be a parametrizationas in Proposition 61 Then the following statements are true
(1) The elements 1205771 120577
119898are linearly independent in 119867
(2) For every ℎ isin 119880 cap M one has the direct sum decom-position
Therefore by Proposition 64 for every 1199100isin R119898 there exists
a (up to indistinguishability) unique weak solution to (192)which according to Proposition 39 is also a strong solutionto (192) The uniqueness of strong solutions to (192) is a con-sequence of Proposition 39 andTheorem 48
Now we are ready to formulate and prove ourmain resultof this section
Theorem 68 The following statements are equivalent
(1) The submanifoldM is locally invariant for (176)(2) One has
M sub D (119860) (195)
120590119895
(ℎ) isin 119879ℎM forallℎ isin M 119886119897119897 119895 isin N (196)
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) isin 119879ℎM forallℎ isin M (197)
(3) The operator 119860 is continuous onM and for each ℎ0isin
M there exists a local strong solution 119883 to (176) withsome lifetime 120591 gt 0 such that
119883119905and120591
isin M forall119905 ge 0 P-119886119897119898119900119904119905 119904119906119903119890119897119910 (198)
Proof (1)rArr(2) Let ℎ isin M be arbitrary By Proposition 61and Remark 60 there exist elements 120577
1 120577
119898isin D(119860
lowast
) anda parametrization 120601 119881 rarr 119880 cap M around ℎ such that theinverse 120601
minus1
119880 cap M rarr 119881 is given by 120601minus1
= ⟨120577 ∙⟩ and 120601
has an extension 120601 isin 1198622
119887(R119898
119867) Since the submanifoldM islocally invariant for (176) there exists a local weak solution119883
to (176) with initial condition ℎ and some lifetime 984858 gt 0 suchthat
119883119905and984858
isin M forall119905 ge 0 P-almost surely (199)
Since119880 is an open neighborhood of ℎ there exists 120598 gt 0 suchthat 119861
120598(ℎ) sub 119880 where 119861
120598(ℎ) denotes the open ball
119861120598(ℎ) = 119892 isin 119867
1003817100381710038171003817119892 minus ℎ1003817100381710038171003817 lt 120598 (200)
We define the stopping time
120591 = 984858 and inf 119905 ge 0 119883119905notin 119861
120598(ℎ) (201)
Since the process119883 has continuous sample paths and satisfies1198830= ℎ we have 120591 gt 0 and P-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (202)
22 International Journal of Stochastic Analysis
Defining the R119898-valued process 119884 = ⟨120577 119883⟩ we have P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (203)
Moreover since 119883 is a weak solution to (176) with initialcondition ℎ setting 119910 = ⟨120577 ℎ⟩ isin 119881 we have P-almost surely
119884119905and120591
= ⟨120577 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120577 120590119895
(119883119904)⟩ 119889120573
119895
119904
= ⟨120577 ℎ⟩ + int
119905and120591
0
120572120601120577
(⟨120577 119883119904⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(⟨120577 119883
119904⟩) 119889120573
119895
119904
= 119910 + int
119905and120591
0
120572120601120577
(119884119904) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(204)
showing that 119884 is a local strong solution to (192) with initialcondition 119910 By Itorsquos formula (Theorem 26) we obtain P-almost surely
119883119905and120591
= 120601 (119884119905and120591
)
= ℎ + int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904)))119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(205)
Now let 120585 isin D(119860lowast
) be arbitrary Then we have P-almostsurely
⟨120585 119883119905and120591
⟩
= ⟨120585 ℎ⟩
+ int
119905and120591
0
⟨120585119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 119863120601 (119884119904) 120590
119895
120601120577(119884
119904)⟩119889120573
119895
119904 119905 ge 0
(206)
On the other hand since 119883 is a local weak solution to (176)with initial condition ℎ and lifetime 120591 we have P-almostsurely for all 119905 ge 0 the identity
⟨120585 119883119905and120591
⟩ = ⟨120585 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904)⟩ 119889120573
119895
119904
(207)
Combining (206) and (207) yields up to indistinguishability
119861 + 119872 = 0 (208)
where the processes 119861 and 119872 are defined as
119861119905= int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩) 119889119904
119905 ge 0
119872119905= sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩ 119889120573
119895
119904
119905 ge 0
(209)
The process 119861 + 119872 is a continuous semimartingale withcanonical decomposition (208) Since the canonical decom-position of a continuous semimartingale is unique up toindistinguishability we deduce that 119861 = 119872 = 0 up toindistinguishability Using the Ito isometry (39) we obtain P-almost surely
int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩
+ ⟨120585 120572 (119883119904) minus 119863120601 (119884
showing that119883 is a local strong solution to (44) with lifetime120591
(3)rArr(1) This implication is a direct consequence ofProposition 29
The results from this section are closely related to the exis-tence of finite dimensional realizations that is the existenceof invariantmanifolds for each starting point ℎ
0 andwe point
out the papers [17ndash22] regarding this topic Furthermore wemention thatTheorem 68 has been extended in [23] to SPDEswith jumps
Acknowledgments
The author is grateful to Daniel Gaigall Georg GrafendorferFlorian Modler and Thomas Salfeld for the valuable com-ments and discussions
References
[1] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions CambridgeUniversity Press CambridgeUK 1992
[2] C Prevot and M Rockner A Concise Course on StochasticPartial Differential Equations Springer Berlin Germany 2007
[3] L Gawarecki and V Mandrekar Stochastic Differential Equa-tions in Infinite Dimensions with Applications to StochasticPartial Differential Equations Springer Berlin Germany 2011
[4] S Peszat and J Zabczyk Stochastic Partial Differential Equationswith Levy Noise Cambridge University Press Cambridge UK2007
[5] S Albeverio V Mandrekar and B Rudiger ldquoExistence of mildsolutions for stochastic differential equations and semilinearequations with non-Gaussian Levy noiserdquo Stochastic Processesand Their Applications vol 119 no 3 pp 835ndash863 2009
[6] D Filipovic S Tappe and J Teichmann ldquoJump-diffusions inHilbert spaces existence stability and numericsrdquo Stochasticsvol 82 no 5 pp 475ndash520 2010
[7] C Marinelli C Prevot and M Rockner ldquoRegular dependenceon initial data for stochastic evolution equations with multi-plicative Poisson noiserdquo Journal of Functional Analysis vol 258no 2 pp 616ndash649 2010
[8] S Tappe ldquoSome refinements of existence results for SPDEsdriven by Wiener processes and Poisson random measuresrdquoInternational Journal of Stochastic Analysis vol 2012 Article ID236327 24 pages 2012
[9] T Nakayama ldquoSupport theorem for mild solutions of SDErsquos inHilbert spacesrdquo Journal of Mathematical Sciences the Universityof Tokyo vol 11 no 3 pp 245ndash311 2004
[10] T Nakayama ldquoViability theorem for SPDErsquos including HJMframeworkrdquo Journal of Mathematical Sciences the University ofTokyo vol 11 no 3 pp 313ndash324 2004
[11] W Rudin Functional Analysis McGraw-Hill New York NYUSA 2nd edition 1991
[12] DWerner Funktionalanalysis Springer Berlin Germany 2007[13] C D Aliprantis and K C Border Infinite Dimensional Analysis
Springer Berlin Germany 3rd edition 2006[14] S N Ethier and T G KurtzMarkov Processes Characterization
andConvergenceWiley Series in Probability and Statistics JohnWiley amp Sons New Jersey NJ USA 1986
[15] V Mandrekar and B Rudiger ldquoExistence and uniqueness ofpath wise solutions for stochastic integral equations driven byLevy noise on separable Banach spacesrdquo Stochastics vol 78 no4 pp 189ndash212 2006
[16] D Filipovic Consistency Problems for Heath-Jarrow-MortonInterest Rate Models Springer Berlin Germany 2001
[17] T Bjork and L Svensson ldquoOn the existence of finite-dimensional realizations for nonlinear forward rate modelsrdquoMathematical Finance vol 11 no 2 pp 205ndash243 2001
[18] T Bjork and C Landen ldquoOn the construction of finite dimen-sional realizations for nonlinear forward rate modelsrdquo Financeand Stochastics vol 6 no 3 pp 303ndash331 2002
[19] D Filipovic and J Teichmann ldquoExistence of invariantmanifoldsfor stochastic equations in infinite dimensionrdquo Journal ofFunctional Analysis vol 197 no 2 pp 398ndash432 2003
[20] D Filipovic and J Teichmann ldquoOn the geometry of the termstructure of interest ratesrdquo Proceedings of The Royal Society ofLondon A vol 460 no 2041 pp 129ndash167 2004
[21] S Tappe ldquoAn alternative approach on the existence of affinerealizations for HJM term structure modelsrdquo Proceedings of
International Journal of Stochastic Analysis 25
TheRoyal Society of LondonA vol 466 no 2122 pp 3033ndash30602010
[22] S Tappe ldquoExistence of affine realizations for Levy term struc-ture modelsrdquo Proceedings of The Royal Society of London A vol468 no 2147 pp 3685ndash3704 2012
[23] D Filipovic S Tappe and J Teichmann ldquoInvariant manifoldswith boundary for jump-diffusionsrdquo httparxivorgabs12021076
) is dense in119867 and hencewe obtain P-almost surely
119883119905and120591
= ℎ0+ int
119905and120591
0
(119860119883119904+ 120572 (119904 119883
119904)) 119889119904
+ int
119905and120591
0
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(53)
Consequently the process 119883 is also a local strong solution to(44) with lifetime 120591
Corollary 31 Let M sub D(119860) be a subset such that 119860 iscontinuous on M and let 119883 be a local weak solution to (44)with lifetime 120591 such that
119883119905and120591
isin M forall119905 ge 0 P-119886119897119898119900119904119905 119904119906119903119890119897119910 (54)
Then 119883 is also a local strong solution to (44) with lifetime 120591
Proof Since M sub D(119860) condition (54) implies that (46) isfulfilled Moreover by the continuity of 119860 onM the samplepaths of the process 119860119883 are P-almost surely continuous andhence we obtain (47) Consequently using Proposition 30the process 119883 is also a local strong solution to (44) withlifetime 120591
Proposition 32 Every strong solution119883 to (44) is also a mildsolution to (44)
Proof According to Lemma 8 the domain (D(119860) sdot D(119860))
endowed with the graph norm is a separable Hilbert spacetoo Hence by Lemma 18 for all 119905 ge 0 the function
119891 [0 119905] times D (119860) 997888rarr 119867 119891 (119904 119909) = 119878119905minus119904
119909 (55)
belongs to the class 11986212loc119887
([0 119905] times D(119860)119867) with partialderivatives
Since by Proposition 14 the domain D((119860lowast
)2
) is dense in(D(119860
lowast
) sdot D(119860lowast)) we get P-almost surely for all 120577 isin D(119860
lowast
)
the identity
⟨120577 119883119905⟩ = ⟨120577 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904⟩
(70)
Since by Proposition 14 the domainD(119860lowast
) is dense in119867 weobtain P-almost surely
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 (71)
proving that 119883 is a mild solution to (44)
Remark 36 Now the proof of Proposition 32 is an immediateconsequence of Propositions 29 and 35
We have just seen that every weak solution to (44) is also amild solution Under the following regularity condition (72)the converse of this statement holds true as well
Proposition 37 Let 119883 be a mild solution to (44) such that
120572 (119904 119883119904) minus 120572 (119904 119883
119904)⟩ 119889119904
= ⟨120577 int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904⟩ minus int
119905
0
⟨120577 120572 (119904 119883119904)⟩119889119904
(74)
Due to assumption (72) we may use Fubinirsquos theorem forstochastic integrals (see [3 Theorem 28]) which togetherwith Lemma 18 gives us P-almost surely
Due to assumption (80) we may use Fubinirsquos theorem forstochastic integrals (see [3 Theorem 28]) which togetherwith Lemma 18 gives us P-almost surely
int
119905
0
(119878119905minus119904
120590 (119904 119883119904) minus 120590 (119904 119883
This proves that 119883 is also a strong solution to (44)
The following result shows that for norm continuoussemigroups the concepts of strong weak and mild solutionsare equivalent In particular this applies for finite dimen-sional state spaces
Proposition 39 Suppose that the semigroup (119878119905)119905ge0
is normcontinuous Let 119883 be a stochastic process with 119883
0= ℎ
0 Then
the following statements are equivalent
(1) The process 119883 is a strong solution to (44)
(2) The process 119883 is a weak solution to (44)
(3) The process 119883 is a mild solution to (44)
Proof (1)rArr(2) This implication is a consequence ofProposition 29
(2)rArr(3) This implication is a consequence ofProposition 35
(3)rArr(1) By Proposition 20 we have 119860 isin 119871(119867) and 119878119905=
119890119905119860 119905 ge 0 Furthermore the family (119890
119905119860
)119905isinR is a 119862
0-group on
119867 Therefore and since 119883 is a mild solution to (44) we haveP-almost surely
(2) An element 119909 isin 119864 is called a fixed point of Φ if onehas
Φ (119909) = 119909 (109)
The following result is the well-known Banach fixed pointtheorem Its proof can be found for example in [13Theorem348]
Theorem 45 (The Banach fixed point theorem) Let 119864 be acomplete metric space and let Φ 119864 rarr 119864 be a contractionThen the mapping Φ has a unique fixed point
In this text we will use the following slight extension ofthe Banach fixed point theorem
Corollary 46 Let 119864 be a complete metric space and let Φ
119864 rarr 119864 be a mapping such that for some 119899 isin N the mappingΦ119899 is a contraction Then the mapping Φ has a unique fixed
point
Proof According to the Banach fixed point theorem(Theorem 45) the mapping Φ
119899 has a unique fixed point thatis there exists a unique element 119909 isin 119864 such that Φ119899
(119909) = 119909Therefore we have
Φ (119909) = Φ (Φ119899
(119909)) = Φ119899
(Φ (119909)) (110)
showing thatΦ(119909) is a fixedpoint ofΦ119899 SinceΦ119899 has a uniquefixed point we deduce thatΦ(119909) = 119909 showing that119909 is a fixedpoint of Φ
In order to prove uniqueness let 119910 isin 119864 be another fixedpoint ofΦ that is we haveΦ(119910) = 119910 By induction we obtain
showing that 119910 is a fixed point of Φ119899 Since the mapping Φ119899
has exactly one fixed point we obtain 119909 = 119910
An indispensable tool for proving uniqueness of mildsolutions to (44) will be the following version of Gronwallrsquosinequality see for example [14 Theorem 51]
Lemma 47 (Gronwallrsquos inequality) Let 119879 ge 0 be fixed let119891 [0 119879] rarr R
and hence by the Cauchy-Schwarz inequality the Ito isom-etry (39) the growth estimate (9) from Lemma 10 and thelocal Lipschitz conditions (113) we obtain
Using Gronwallrsquos inequality (see Lemma 47) we deduce that119891 equiv 0 Thus by the continuity of the sample paths of 119883 and119884 we obtain
P(⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ) = 1 forall119899 isin N (120)
and hence by the continuity of the probability measureP weconclude that
P(⋂
119905ge0
119883119905and120591
1Γ= 119884
119905and1205911Γ)
= P(⋂
119899isinN
⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ)
= lim119899rarrinfin
P(⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ) = 1
(121)
which completes the proof
The local Lipschitz conditions (113) are in general notsufficient in order to ensure the existence of mild solutions tothe SPDE (44) Now we will prove that the existence of mildsolutions follows from global Lipschitz and linear growthconditions on 120572 and 120590 For this we recall an auxiliary resultwhich extends the Ito isometry (39)
Lemma 49 Let 119879 ge 0 be arbitrary and let 119883 = (119883119905)119905isin[0119879]
for all 119905 ge 0 and all ℎ isin 119867 Then for every F0-measurable
random variable ℎ0
Ω rarr 119867 there exists a (up toindistinguishability) unique mild solution 119883 to (44)
Proof The uniqueness of mild solutions to (44) is a directconsequence of Theorem 48 and hence we may concentrateon the existence proof which we divide into the followingseveral steps
Step 1 First we suppose that the initial condition ℎ0satisfies
E[ℎ02119901
] lt infin for some 119901 gt 1 Let 119879 ge 0 be arbitrary Wedefine the Banach space
1198712119901
119879(119867) = 119871
2119901
(Ω times [0 119879] P119879P otimes 119889119905119867) (129)
and prove that the variation of constants equation
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(130)
has a unique solution in the space 1198712119901
119879(119867) This is done in the
following three steps
Step 11 For 119883 isin 1198712119901
119879(119867) we define the process Φ119883 by
(Φ119883)119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(131)
Then the process Φ119883 is well defined Indeed by the growthestimate (9) the linear growth condition (127) and Holderrsquosinequality we have
Consequently there exists an index 119899 isin N such that Φ119899 is
a contraction and hence according to the extension of theBanach fixed point theorem (see Corollary 46) the mappingΦ has a unique fixed point 119883 isin 119871
2119901
119879(119867) This fixed point 119883 is
a solution to the variation of constants equation (130) Since119879 ge 0 was arbitrary there exists a process 119883 which is a solu-tion of the variation of constants equation
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
119905 ge 0
(146)
Step 14 In order to prove that 119883 is a mild solution to (44)it remains to show that 119883 has a continuous version ByLemma 12 the process
119905 997891997888rarr 119878119905ℎ0 119905 ge 0 (147)
is continuous and by Proposition 41 the process
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 119905 ge 0 (148)
is continuous too Moreover for every 119879 ge 0 we have by thelinear growth condition (128) Holderrsquos inequality and since119883 isin 119871
Thus by Proposition 43 the stochastic convolution 120590 ⋆ 119882
given by
(120590 ⋆ 119882)119905= int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0 (150)
has a continuous version and consequently the process119883hasa continuous version too This continuous version is a mildsolution to (44)
Step 2 Now let ℎ0 Ω rarr 119867 be an arbitraryF
0-measurable
random variable We define the sequence (ℎ119899)119899isinN of F
0-
measurable random variables as
ℎ119899
0= ℎ
01ℎ0le119899
119899 isin N (151)
Let 119899 isin N be arbitrary Then as ℎ119899
0is bounded we have
E[ℎ119899
02119901
] lt infin for all 119901 gt 1 By Step 1 the SPDE
119889119883119899
119905= (119860119883
119899
119905+ 120572 (119905 119883
119899
119905)) 119889119905 + 120590 (119905 119883
119899
119905) 119889119882
119905
119883119899
0= ℎ
119899
0
(152)
18 International Journal of Stochastic Analysis
has amild solution119883119899We define the sequence (Ω
119899)119899isinN sub F
0
as
Ω119899=
1003817100381710038171003817ℎ01003817100381710038171003817 le 119899 119899 isin N (153)
Then we have Ω119899sub Ω
119898for 119899 le 119898 Ω = ⋃
119899isinN Ω119899 and
Ω119899sub ℎ
119899
0= ℎ
119898
0 sub ℎ
119899
0= ℎ
0 forall119899 le 119898 (154)
Thus byTheorem 48 we have (up to indistinguishability)
119883119899
1Ω119899
= 119883119898
1Ω119899
forall119899 le 119898 (155)
Consequently the process
119883 = lim119899rarrinfin
119883119899
1Ω119899
(156)
is a well-defined continuous and adapted process and wehave
119883119899
1Ω119899
= 119883119898
1Ω119899
= 1198831Ω119899
forall119899 le 119898 (157)
Furthermore we obtain P-almost surely
119883119905= lim
119899rarrinfin
119883119899
1199051Ω119899
= lim119899rarrinfin
1Ω119899
(119878119905ℎ119899
0+ int
119905
0
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+int
119905
0
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ119899
0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= lim119899rarrinfin
1Ω119899
(119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= 119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(158)
proving that 119883 is a mild solution to (44)
Remark 51 For the proof of Theorem 50 we have usedCorollary 46 which is a slight extension of the Banach fixedpoint theorem Such an idea has been applied for example in[15]
Remark 52 A recent method for proving existence anduniqueness of mild solutions to the SPDE (44) is the methodof the moving frame presented in [6] see also [8] It allowsto reduce SPDE problems to the study of SDEs in infinitedimension In order to apply this method we need that thesemigroup (119878
119905)119905ge0
is a semigroup of pseudocontractions
We close this section with a consequence about theexistence of weak solutions
Corollary 53 Suppose that conditions (125)ndash(128) are ful-filled Let ℎ
0 Ω rarr 119867 be aF
0-measurable random variable
such that E[ℎ02119901
] lt infin for some 119901 gt 1 Then there exists a(up to indistinguishability) unique weak solution 119883 to (44)
Proof According to Proposition 35 every weak solution119883 to(44) is also a mild solution to (44)Therefore the uniquenessof weak solutions to (44) is a consequence of Theorem 48
It remains to prove the existence of a weak solution to(44) Let 119879 ge 0 be arbitrary By Theorem 50 and its proofthere exists a mild solution119883 isin 119871
2119901
119879(119867) to (44) By the linear
growth condition (128) and Holderrsquos inequality we obtain
showing that condition (72) is fulfilled Thus byProposition 37 the process 119883 is also a weak solution to(44)
8 Invariant Manifolds for WeakSolutions to SPDEs
In this section we deal with invariant manifolds for time-homogeneous SPDEs of the type (44) This topic arises fromthe natural desire to express the solutions of the SPDE (44)which generally live in the infinite dimensional Hilbert space119867 by means of a finite dimensional state process and thusto ensure larger analytical tractability Our goal is to findconditions on the generator 119860 and the coefficients 120572 120590 suchthat for every starting point of a finite dimensional submani-fold the solution process stays on the submanifold
We start with the required preliminaries about finitedimensional submanifolds inHilbert spaces In the sequel let119867 be a separable Hilbert space
Definition 54 Let 119898 119896 isin N be positive integers A subsetM sub 119867 is called an 119898-dimensional 119862119896-submanifold of 119867
International Journal of Stochastic Analysis 19
if for every ℎ isin M there exist an open neighborhood 119880 sub 119867
of ℎ an open set 119881 sub R119898 and a mapping 120601 isin 1198622
(119881119867) suchthat
(1) the mapping 120601 119881 rarr 119880 cap M is a homeomorphism(2) for all 119910 isin 119881 the mapping 119863120601(119910) is injective
The mapping 120601 is called a parametrization ofM around ℎ
In what follows let M be an 119898-dimensional 119862119896-
submanifold of 119867
Lemma 55 Let 120601119894
119881119894
rarr 119880119894cap M 119894 = 1 2 be two
parametrizations with119882= 1198801cap119880
2capM = 0Then themapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (160)
is a 119862119896-diffeomorphism
Proof See [16 Lemma 611]
Corollary 56 Let ℎ isin M be arbitrary and let 120601119894
119881119894
rarr
119880119894capM 119894 = 1 2 be two parametrizations ofM around ℎ Then
one has
1198631206011(119910
1) (R
119898
) = 1198631206012(119910
2) (R
119898
) (161)
where 119910119894= 120601
minus1
119894(ℎ) for 119894 = 1 2
Proof Since1198801and119880
2are open neighborhoods of ℎ we have
119882= 1198801cap 119880
2cap M = 0 Thus by Lemma 55 the mapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (162)
is a 119862119896-diffeomorphism Using the chain rule we obtain
1198631206012(119910
2) (R
119898
)
= 119863 (1206011∘ (120601
minus1
1∘ 120601
2)) (119910
2) (R
119898
)
= 1198631206011(119910
1)119863 (120601
minus1
1∘ 120601
2) (119910
2) (R
119898
)
sub 1198631206011(119910
1) (R
119898
)
(163)
and analogously we prove that 1198631206011(119910
1)(R119898
) sub
1198631206012(119910
2)(R119898
)
Definition 57 Let ℎ isin M be arbitraryThe tangent space ofMto ℎ is the subspace
119879ℎM = 119863120601 (119910) (R
119898
) (164)
where 119910 = 120601minus1
(ℎ) and 120601 119881 rarr 119880 cap M denotes aparametrization ofM around ℎ
Remark 58 Note that according to Corollary 56 theDefinition 57 of the tangent space 119879
ℎM does not depend on
the choice of the parametrization 120601 119881 rarr 119880 cap M
Proposition 59 Let ℎ isin M be arbitrary and let 120601 119881 rarr
119880 cap M be a parametrization ofM around ℎ Then there existan open set 119881
0sub 119881 an open neighborhood 119880
0sub 119880 of ℎ and
a mapping 120601 isin 119862119896
119887(R119898
119867) with 120601|1198810
= 120601|1198810
such that 120601|1198810
1198810
rarr 1198800cap M is a parametrization ofM around ℎ too
Proof See [16 Remark 611]
Remark 60 By Proposition 59 we may assume that anyparametrization 120601 119881 rarr 119880 cap M has an extension 120601 isin
119862119896
119887(R119898
119867)
Proposition 61 Let 119863 sub 119867 be a dense subset For every ℎ0isin
M there exist 1205771 120577
119898isin 119863 and a parametrization 120601 119881 rarr
119880 cap M around ℎ0such that
120601 (⟨120577 ℎ⟩) = ℎ forallℎ isin 119880 cap M (165)
where one uses the notation ⟨120577 ℎ⟩ = (⟨1205771 ℎ⟩ ⟨120577
119898 ℎ⟩) isin
R119898
Proof See [16 Proposition 612]
Proposition 62 Let 120601 119881 rarr 119880 cap M be a parametrizationas in Proposition 61 Then the following statements are true
(1) The elements 1205771 120577
119898are linearly independent in 119867
(2) For every ℎ isin 119880 cap M one has the direct sum decom-position
Therefore by Proposition 64 for every 1199100isin R119898 there exists
a (up to indistinguishability) unique weak solution to (192)which according to Proposition 39 is also a strong solutionto (192) The uniqueness of strong solutions to (192) is a con-sequence of Proposition 39 andTheorem 48
Now we are ready to formulate and prove ourmain resultof this section
Theorem 68 The following statements are equivalent
(1) The submanifoldM is locally invariant for (176)(2) One has
M sub D (119860) (195)
120590119895
(ℎ) isin 119879ℎM forallℎ isin M 119886119897119897 119895 isin N (196)
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) isin 119879ℎM forallℎ isin M (197)
(3) The operator 119860 is continuous onM and for each ℎ0isin
M there exists a local strong solution 119883 to (176) withsome lifetime 120591 gt 0 such that
119883119905and120591
isin M forall119905 ge 0 P-119886119897119898119900119904119905 119904119906119903119890119897119910 (198)
Proof (1)rArr(2) Let ℎ isin M be arbitrary By Proposition 61and Remark 60 there exist elements 120577
1 120577
119898isin D(119860
lowast
) anda parametrization 120601 119881 rarr 119880 cap M around ℎ such that theinverse 120601
minus1
119880 cap M rarr 119881 is given by 120601minus1
= ⟨120577 ∙⟩ and 120601
has an extension 120601 isin 1198622
119887(R119898
119867) Since the submanifoldM islocally invariant for (176) there exists a local weak solution119883
to (176) with initial condition ℎ and some lifetime 984858 gt 0 suchthat
119883119905and984858
isin M forall119905 ge 0 P-almost surely (199)
Since119880 is an open neighborhood of ℎ there exists 120598 gt 0 suchthat 119861
120598(ℎ) sub 119880 where 119861
120598(ℎ) denotes the open ball
119861120598(ℎ) = 119892 isin 119867
1003817100381710038171003817119892 minus ℎ1003817100381710038171003817 lt 120598 (200)
We define the stopping time
120591 = 984858 and inf 119905 ge 0 119883119905notin 119861
120598(ℎ) (201)
Since the process119883 has continuous sample paths and satisfies1198830= ℎ we have 120591 gt 0 and P-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (202)
22 International Journal of Stochastic Analysis
Defining the R119898-valued process 119884 = ⟨120577 119883⟩ we have P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (203)
Moreover since 119883 is a weak solution to (176) with initialcondition ℎ setting 119910 = ⟨120577 ℎ⟩ isin 119881 we have P-almost surely
119884119905and120591
= ⟨120577 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120577 120590119895
(119883119904)⟩ 119889120573
119895
119904
= ⟨120577 ℎ⟩ + int
119905and120591
0
120572120601120577
(⟨120577 119883119904⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(⟨120577 119883
119904⟩) 119889120573
119895
119904
= 119910 + int
119905and120591
0
120572120601120577
(119884119904) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(204)
showing that 119884 is a local strong solution to (192) with initialcondition 119910 By Itorsquos formula (Theorem 26) we obtain P-almost surely
119883119905and120591
= 120601 (119884119905and120591
)
= ℎ + int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904)))119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(205)
Now let 120585 isin D(119860lowast
) be arbitrary Then we have P-almostsurely
⟨120585 119883119905and120591
⟩
= ⟨120585 ℎ⟩
+ int
119905and120591
0
⟨120585119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 119863120601 (119884119904) 120590
119895
120601120577(119884
119904)⟩119889120573
119895
119904 119905 ge 0
(206)
On the other hand since 119883 is a local weak solution to (176)with initial condition ℎ and lifetime 120591 we have P-almostsurely for all 119905 ge 0 the identity
⟨120585 119883119905and120591
⟩ = ⟨120585 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904)⟩ 119889120573
119895
119904
(207)
Combining (206) and (207) yields up to indistinguishability
119861 + 119872 = 0 (208)
where the processes 119861 and 119872 are defined as
119861119905= int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩) 119889119904
119905 ge 0
119872119905= sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩ 119889120573
119895
119904
119905 ge 0
(209)
The process 119861 + 119872 is a continuous semimartingale withcanonical decomposition (208) Since the canonical decom-position of a continuous semimartingale is unique up toindistinguishability we deduce that 119861 = 119872 = 0 up toindistinguishability Using the Ito isometry (39) we obtain P-almost surely
int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩
+ ⟨120585 120572 (119883119904) minus 119863120601 (119884
showing that119883 is a local strong solution to (44) with lifetime120591
(3)rArr(1) This implication is a direct consequence ofProposition 29
The results from this section are closely related to the exis-tence of finite dimensional realizations that is the existenceof invariantmanifolds for each starting point ℎ
0 andwe point
out the papers [17ndash22] regarding this topic Furthermore wemention thatTheorem 68 has been extended in [23] to SPDEswith jumps
Acknowledgments
The author is grateful to Daniel Gaigall Georg GrafendorferFlorian Modler and Thomas Salfeld for the valuable com-ments and discussions
References
[1] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions CambridgeUniversity Press CambridgeUK 1992
[2] C Prevot and M Rockner A Concise Course on StochasticPartial Differential Equations Springer Berlin Germany 2007
[3] L Gawarecki and V Mandrekar Stochastic Differential Equa-tions in Infinite Dimensions with Applications to StochasticPartial Differential Equations Springer Berlin Germany 2011
[4] S Peszat and J Zabczyk Stochastic Partial Differential Equationswith Levy Noise Cambridge University Press Cambridge UK2007
[5] S Albeverio V Mandrekar and B Rudiger ldquoExistence of mildsolutions for stochastic differential equations and semilinearequations with non-Gaussian Levy noiserdquo Stochastic Processesand Their Applications vol 119 no 3 pp 835ndash863 2009
[6] D Filipovic S Tappe and J Teichmann ldquoJump-diffusions inHilbert spaces existence stability and numericsrdquo Stochasticsvol 82 no 5 pp 475ndash520 2010
[7] C Marinelli C Prevot and M Rockner ldquoRegular dependenceon initial data for stochastic evolution equations with multi-plicative Poisson noiserdquo Journal of Functional Analysis vol 258no 2 pp 616ndash649 2010
[8] S Tappe ldquoSome refinements of existence results for SPDEsdriven by Wiener processes and Poisson random measuresrdquoInternational Journal of Stochastic Analysis vol 2012 Article ID236327 24 pages 2012
[9] T Nakayama ldquoSupport theorem for mild solutions of SDErsquos inHilbert spacesrdquo Journal of Mathematical Sciences the Universityof Tokyo vol 11 no 3 pp 245ndash311 2004
[10] T Nakayama ldquoViability theorem for SPDErsquos including HJMframeworkrdquo Journal of Mathematical Sciences the University ofTokyo vol 11 no 3 pp 313ndash324 2004
[11] W Rudin Functional Analysis McGraw-Hill New York NYUSA 2nd edition 1991
[12] DWerner Funktionalanalysis Springer Berlin Germany 2007[13] C D Aliprantis and K C Border Infinite Dimensional Analysis
Springer Berlin Germany 3rd edition 2006[14] S N Ethier and T G KurtzMarkov Processes Characterization
andConvergenceWiley Series in Probability and Statistics JohnWiley amp Sons New Jersey NJ USA 1986
[15] V Mandrekar and B Rudiger ldquoExistence and uniqueness ofpath wise solutions for stochastic integral equations driven byLevy noise on separable Banach spacesrdquo Stochastics vol 78 no4 pp 189ndash212 2006
[16] D Filipovic Consistency Problems for Heath-Jarrow-MortonInterest Rate Models Springer Berlin Germany 2001
[17] T Bjork and L Svensson ldquoOn the existence of finite-dimensional realizations for nonlinear forward rate modelsrdquoMathematical Finance vol 11 no 2 pp 205ndash243 2001
[18] T Bjork and C Landen ldquoOn the construction of finite dimen-sional realizations for nonlinear forward rate modelsrdquo Financeand Stochastics vol 6 no 3 pp 303ndash331 2002
[19] D Filipovic and J Teichmann ldquoExistence of invariantmanifoldsfor stochastic equations in infinite dimensionrdquo Journal ofFunctional Analysis vol 197 no 2 pp 398ndash432 2003
[20] D Filipovic and J Teichmann ldquoOn the geometry of the termstructure of interest ratesrdquo Proceedings of The Royal Society ofLondon A vol 460 no 2041 pp 129ndash167 2004
[21] S Tappe ldquoAn alternative approach on the existence of affinerealizations for HJM term structure modelsrdquo Proceedings of
International Journal of Stochastic Analysis 25
TheRoyal Society of LondonA vol 466 no 2122 pp 3033ndash30602010
[22] S Tappe ldquoExistence of affine realizations for Levy term struc-ture modelsrdquo Proceedings of The Royal Society of London A vol468 no 2147 pp 3685ndash3704 2012
[23] D Filipovic S Tappe and J Teichmann ldquoInvariant manifoldswith boundary for jump-diffusionsrdquo httparxivorgabs12021076
Since by Proposition 14 the domain D((119860lowast
)2
) is dense in(D(119860
lowast
) sdot D(119860lowast)) we get P-almost surely for all 120577 isin D(119860
lowast
)
the identity
⟨120577 119883119905⟩ = ⟨120577 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904⟩
(70)
Since by Proposition 14 the domainD(119860lowast
) is dense in119867 weobtain P-almost surely
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 (71)
proving that 119883 is a mild solution to (44)
Remark 36 Now the proof of Proposition 32 is an immediateconsequence of Propositions 29 and 35
We have just seen that every weak solution to (44) is also amild solution Under the following regularity condition (72)the converse of this statement holds true as well
Proposition 37 Let 119883 be a mild solution to (44) such that
120572 (119904 119883119904) minus 120572 (119904 119883
119904)⟩ 119889119904
= ⟨120577 int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904⟩ minus int
119905
0
⟨120577 120572 (119904 119883119904)⟩119889119904
(74)
Due to assumption (72) we may use Fubinirsquos theorem forstochastic integrals (see [3 Theorem 28]) which togetherwith Lemma 18 gives us P-almost surely
Due to assumption (80) we may use Fubinirsquos theorem forstochastic integrals (see [3 Theorem 28]) which togetherwith Lemma 18 gives us P-almost surely
int
119905
0
(119878119905minus119904
120590 (119904 119883119904) minus 120590 (119904 119883
This proves that 119883 is also a strong solution to (44)
The following result shows that for norm continuoussemigroups the concepts of strong weak and mild solutionsare equivalent In particular this applies for finite dimen-sional state spaces
Proposition 39 Suppose that the semigroup (119878119905)119905ge0
is normcontinuous Let 119883 be a stochastic process with 119883
0= ℎ
0 Then
the following statements are equivalent
(1) The process 119883 is a strong solution to (44)
(2) The process 119883 is a weak solution to (44)
(3) The process 119883 is a mild solution to (44)
Proof (1)rArr(2) This implication is a consequence ofProposition 29
(2)rArr(3) This implication is a consequence ofProposition 35
(3)rArr(1) By Proposition 20 we have 119860 isin 119871(119867) and 119878119905=
119890119905119860 119905 ge 0 Furthermore the family (119890
119905119860
)119905isinR is a 119862
0-group on
119867 Therefore and since 119883 is a mild solution to (44) we haveP-almost surely
(2) An element 119909 isin 119864 is called a fixed point of Φ if onehas
Φ (119909) = 119909 (109)
The following result is the well-known Banach fixed pointtheorem Its proof can be found for example in [13Theorem348]
Theorem 45 (The Banach fixed point theorem) Let 119864 be acomplete metric space and let Φ 119864 rarr 119864 be a contractionThen the mapping Φ has a unique fixed point
In this text we will use the following slight extension ofthe Banach fixed point theorem
Corollary 46 Let 119864 be a complete metric space and let Φ
119864 rarr 119864 be a mapping such that for some 119899 isin N the mappingΦ119899 is a contraction Then the mapping Φ has a unique fixed
point
Proof According to the Banach fixed point theorem(Theorem 45) the mapping Φ
119899 has a unique fixed point thatis there exists a unique element 119909 isin 119864 such that Φ119899
(119909) = 119909Therefore we have
Φ (119909) = Φ (Φ119899
(119909)) = Φ119899
(Φ (119909)) (110)
showing thatΦ(119909) is a fixedpoint ofΦ119899 SinceΦ119899 has a uniquefixed point we deduce thatΦ(119909) = 119909 showing that119909 is a fixedpoint of Φ
In order to prove uniqueness let 119910 isin 119864 be another fixedpoint ofΦ that is we haveΦ(119910) = 119910 By induction we obtain
showing that 119910 is a fixed point of Φ119899 Since the mapping Φ119899
has exactly one fixed point we obtain 119909 = 119910
An indispensable tool for proving uniqueness of mildsolutions to (44) will be the following version of Gronwallrsquosinequality see for example [14 Theorem 51]
Lemma 47 (Gronwallrsquos inequality) Let 119879 ge 0 be fixed let119891 [0 119879] rarr R
and hence by the Cauchy-Schwarz inequality the Ito isom-etry (39) the growth estimate (9) from Lemma 10 and thelocal Lipschitz conditions (113) we obtain
Using Gronwallrsquos inequality (see Lemma 47) we deduce that119891 equiv 0 Thus by the continuity of the sample paths of 119883 and119884 we obtain
P(⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ) = 1 forall119899 isin N (120)
and hence by the continuity of the probability measureP weconclude that
P(⋂
119905ge0
119883119905and120591
1Γ= 119884
119905and1205911Γ)
= P(⋂
119899isinN
⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ)
= lim119899rarrinfin
P(⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ) = 1
(121)
which completes the proof
The local Lipschitz conditions (113) are in general notsufficient in order to ensure the existence of mild solutions tothe SPDE (44) Now we will prove that the existence of mildsolutions follows from global Lipschitz and linear growthconditions on 120572 and 120590 For this we recall an auxiliary resultwhich extends the Ito isometry (39)
Lemma 49 Let 119879 ge 0 be arbitrary and let 119883 = (119883119905)119905isin[0119879]
for all 119905 ge 0 and all ℎ isin 119867 Then for every F0-measurable
random variable ℎ0
Ω rarr 119867 there exists a (up toindistinguishability) unique mild solution 119883 to (44)
Proof The uniqueness of mild solutions to (44) is a directconsequence of Theorem 48 and hence we may concentrateon the existence proof which we divide into the followingseveral steps
Step 1 First we suppose that the initial condition ℎ0satisfies
E[ℎ02119901
] lt infin for some 119901 gt 1 Let 119879 ge 0 be arbitrary Wedefine the Banach space
1198712119901
119879(119867) = 119871
2119901
(Ω times [0 119879] P119879P otimes 119889119905119867) (129)
and prove that the variation of constants equation
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(130)
has a unique solution in the space 1198712119901
119879(119867) This is done in the
following three steps
Step 11 For 119883 isin 1198712119901
119879(119867) we define the process Φ119883 by
(Φ119883)119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(131)
Then the process Φ119883 is well defined Indeed by the growthestimate (9) the linear growth condition (127) and Holderrsquosinequality we have
Consequently there exists an index 119899 isin N such that Φ119899 is
a contraction and hence according to the extension of theBanach fixed point theorem (see Corollary 46) the mappingΦ has a unique fixed point 119883 isin 119871
2119901
119879(119867) This fixed point 119883 is
a solution to the variation of constants equation (130) Since119879 ge 0 was arbitrary there exists a process 119883 which is a solu-tion of the variation of constants equation
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
119905 ge 0
(146)
Step 14 In order to prove that 119883 is a mild solution to (44)it remains to show that 119883 has a continuous version ByLemma 12 the process
119905 997891997888rarr 119878119905ℎ0 119905 ge 0 (147)
is continuous and by Proposition 41 the process
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 119905 ge 0 (148)
is continuous too Moreover for every 119879 ge 0 we have by thelinear growth condition (128) Holderrsquos inequality and since119883 isin 119871
Thus by Proposition 43 the stochastic convolution 120590 ⋆ 119882
given by
(120590 ⋆ 119882)119905= int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0 (150)
has a continuous version and consequently the process119883hasa continuous version too This continuous version is a mildsolution to (44)
Step 2 Now let ℎ0 Ω rarr 119867 be an arbitraryF
0-measurable
random variable We define the sequence (ℎ119899)119899isinN of F
0-
measurable random variables as
ℎ119899
0= ℎ
01ℎ0le119899
119899 isin N (151)
Let 119899 isin N be arbitrary Then as ℎ119899
0is bounded we have
E[ℎ119899
02119901
] lt infin for all 119901 gt 1 By Step 1 the SPDE
119889119883119899
119905= (119860119883
119899
119905+ 120572 (119905 119883
119899
119905)) 119889119905 + 120590 (119905 119883
119899
119905) 119889119882
119905
119883119899
0= ℎ
119899
0
(152)
18 International Journal of Stochastic Analysis
has amild solution119883119899We define the sequence (Ω
119899)119899isinN sub F
0
as
Ω119899=
1003817100381710038171003817ℎ01003817100381710038171003817 le 119899 119899 isin N (153)
Then we have Ω119899sub Ω
119898for 119899 le 119898 Ω = ⋃
119899isinN Ω119899 and
Ω119899sub ℎ
119899
0= ℎ
119898
0 sub ℎ
119899
0= ℎ
0 forall119899 le 119898 (154)
Thus byTheorem 48 we have (up to indistinguishability)
119883119899
1Ω119899
= 119883119898
1Ω119899
forall119899 le 119898 (155)
Consequently the process
119883 = lim119899rarrinfin
119883119899
1Ω119899
(156)
is a well-defined continuous and adapted process and wehave
119883119899
1Ω119899
= 119883119898
1Ω119899
= 1198831Ω119899
forall119899 le 119898 (157)
Furthermore we obtain P-almost surely
119883119905= lim
119899rarrinfin
119883119899
1199051Ω119899
= lim119899rarrinfin
1Ω119899
(119878119905ℎ119899
0+ int
119905
0
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+int
119905
0
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ119899
0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= lim119899rarrinfin
1Ω119899
(119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= 119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(158)
proving that 119883 is a mild solution to (44)
Remark 51 For the proof of Theorem 50 we have usedCorollary 46 which is a slight extension of the Banach fixedpoint theorem Such an idea has been applied for example in[15]
Remark 52 A recent method for proving existence anduniqueness of mild solutions to the SPDE (44) is the methodof the moving frame presented in [6] see also [8] It allowsto reduce SPDE problems to the study of SDEs in infinitedimension In order to apply this method we need that thesemigroup (119878
119905)119905ge0
is a semigroup of pseudocontractions
We close this section with a consequence about theexistence of weak solutions
Corollary 53 Suppose that conditions (125)ndash(128) are ful-filled Let ℎ
0 Ω rarr 119867 be aF
0-measurable random variable
such that E[ℎ02119901
] lt infin for some 119901 gt 1 Then there exists a(up to indistinguishability) unique weak solution 119883 to (44)
Proof According to Proposition 35 every weak solution119883 to(44) is also a mild solution to (44)Therefore the uniquenessof weak solutions to (44) is a consequence of Theorem 48
It remains to prove the existence of a weak solution to(44) Let 119879 ge 0 be arbitrary By Theorem 50 and its proofthere exists a mild solution119883 isin 119871
2119901
119879(119867) to (44) By the linear
growth condition (128) and Holderrsquos inequality we obtain
showing that condition (72) is fulfilled Thus byProposition 37 the process 119883 is also a weak solution to(44)
8 Invariant Manifolds for WeakSolutions to SPDEs
In this section we deal with invariant manifolds for time-homogeneous SPDEs of the type (44) This topic arises fromthe natural desire to express the solutions of the SPDE (44)which generally live in the infinite dimensional Hilbert space119867 by means of a finite dimensional state process and thusto ensure larger analytical tractability Our goal is to findconditions on the generator 119860 and the coefficients 120572 120590 suchthat for every starting point of a finite dimensional submani-fold the solution process stays on the submanifold
We start with the required preliminaries about finitedimensional submanifolds inHilbert spaces In the sequel let119867 be a separable Hilbert space
Definition 54 Let 119898 119896 isin N be positive integers A subsetM sub 119867 is called an 119898-dimensional 119862119896-submanifold of 119867
International Journal of Stochastic Analysis 19
if for every ℎ isin M there exist an open neighborhood 119880 sub 119867
of ℎ an open set 119881 sub R119898 and a mapping 120601 isin 1198622
(119881119867) suchthat
(1) the mapping 120601 119881 rarr 119880 cap M is a homeomorphism(2) for all 119910 isin 119881 the mapping 119863120601(119910) is injective
The mapping 120601 is called a parametrization ofM around ℎ
In what follows let M be an 119898-dimensional 119862119896-
submanifold of 119867
Lemma 55 Let 120601119894
119881119894
rarr 119880119894cap M 119894 = 1 2 be two
parametrizations with119882= 1198801cap119880
2capM = 0Then themapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (160)
is a 119862119896-diffeomorphism
Proof See [16 Lemma 611]
Corollary 56 Let ℎ isin M be arbitrary and let 120601119894
119881119894
rarr
119880119894capM 119894 = 1 2 be two parametrizations ofM around ℎ Then
one has
1198631206011(119910
1) (R
119898
) = 1198631206012(119910
2) (R
119898
) (161)
where 119910119894= 120601
minus1
119894(ℎ) for 119894 = 1 2
Proof Since1198801and119880
2are open neighborhoods of ℎ we have
119882= 1198801cap 119880
2cap M = 0 Thus by Lemma 55 the mapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (162)
is a 119862119896-diffeomorphism Using the chain rule we obtain
1198631206012(119910
2) (R
119898
)
= 119863 (1206011∘ (120601
minus1
1∘ 120601
2)) (119910
2) (R
119898
)
= 1198631206011(119910
1)119863 (120601
minus1
1∘ 120601
2) (119910
2) (R
119898
)
sub 1198631206011(119910
1) (R
119898
)
(163)
and analogously we prove that 1198631206011(119910
1)(R119898
) sub
1198631206012(119910
2)(R119898
)
Definition 57 Let ℎ isin M be arbitraryThe tangent space ofMto ℎ is the subspace
119879ℎM = 119863120601 (119910) (R
119898
) (164)
where 119910 = 120601minus1
(ℎ) and 120601 119881 rarr 119880 cap M denotes aparametrization ofM around ℎ
Remark 58 Note that according to Corollary 56 theDefinition 57 of the tangent space 119879
ℎM does not depend on
the choice of the parametrization 120601 119881 rarr 119880 cap M
Proposition 59 Let ℎ isin M be arbitrary and let 120601 119881 rarr
119880 cap M be a parametrization ofM around ℎ Then there existan open set 119881
0sub 119881 an open neighborhood 119880
0sub 119880 of ℎ and
a mapping 120601 isin 119862119896
119887(R119898
119867) with 120601|1198810
= 120601|1198810
such that 120601|1198810
1198810
rarr 1198800cap M is a parametrization ofM around ℎ too
Proof See [16 Remark 611]
Remark 60 By Proposition 59 we may assume that anyparametrization 120601 119881 rarr 119880 cap M has an extension 120601 isin
119862119896
119887(R119898
119867)
Proposition 61 Let 119863 sub 119867 be a dense subset For every ℎ0isin
M there exist 1205771 120577
119898isin 119863 and a parametrization 120601 119881 rarr
119880 cap M around ℎ0such that
120601 (⟨120577 ℎ⟩) = ℎ forallℎ isin 119880 cap M (165)
where one uses the notation ⟨120577 ℎ⟩ = (⟨1205771 ℎ⟩ ⟨120577
119898 ℎ⟩) isin
R119898
Proof See [16 Proposition 612]
Proposition 62 Let 120601 119881 rarr 119880 cap M be a parametrizationas in Proposition 61 Then the following statements are true
(1) The elements 1205771 120577
119898are linearly independent in 119867
(2) For every ℎ isin 119880 cap M one has the direct sum decom-position
Therefore by Proposition 64 for every 1199100isin R119898 there exists
a (up to indistinguishability) unique weak solution to (192)which according to Proposition 39 is also a strong solutionto (192) The uniqueness of strong solutions to (192) is a con-sequence of Proposition 39 andTheorem 48
Now we are ready to formulate and prove ourmain resultof this section
Theorem 68 The following statements are equivalent
(1) The submanifoldM is locally invariant for (176)(2) One has
M sub D (119860) (195)
120590119895
(ℎ) isin 119879ℎM forallℎ isin M 119886119897119897 119895 isin N (196)
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) isin 119879ℎM forallℎ isin M (197)
(3) The operator 119860 is continuous onM and for each ℎ0isin
M there exists a local strong solution 119883 to (176) withsome lifetime 120591 gt 0 such that
119883119905and120591
isin M forall119905 ge 0 P-119886119897119898119900119904119905 119904119906119903119890119897119910 (198)
Proof (1)rArr(2) Let ℎ isin M be arbitrary By Proposition 61and Remark 60 there exist elements 120577
1 120577
119898isin D(119860
lowast
) anda parametrization 120601 119881 rarr 119880 cap M around ℎ such that theinverse 120601
minus1
119880 cap M rarr 119881 is given by 120601minus1
= ⟨120577 ∙⟩ and 120601
has an extension 120601 isin 1198622
119887(R119898
119867) Since the submanifoldM islocally invariant for (176) there exists a local weak solution119883
to (176) with initial condition ℎ and some lifetime 984858 gt 0 suchthat
119883119905and984858
isin M forall119905 ge 0 P-almost surely (199)
Since119880 is an open neighborhood of ℎ there exists 120598 gt 0 suchthat 119861
120598(ℎ) sub 119880 where 119861
120598(ℎ) denotes the open ball
119861120598(ℎ) = 119892 isin 119867
1003817100381710038171003817119892 minus ℎ1003817100381710038171003817 lt 120598 (200)
We define the stopping time
120591 = 984858 and inf 119905 ge 0 119883119905notin 119861
120598(ℎ) (201)
Since the process119883 has continuous sample paths and satisfies1198830= ℎ we have 120591 gt 0 and P-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (202)
22 International Journal of Stochastic Analysis
Defining the R119898-valued process 119884 = ⟨120577 119883⟩ we have P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (203)
Moreover since 119883 is a weak solution to (176) with initialcondition ℎ setting 119910 = ⟨120577 ℎ⟩ isin 119881 we have P-almost surely
119884119905and120591
= ⟨120577 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120577 120590119895
(119883119904)⟩ 119889120573
119895
119904
= ⟨120577 ℎ⟩ + int
119905and120591
0
120572120601120577
(⟨120577 119883119904⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(⟨120577 119883
119904⟩) 119889120573
119895
119904
= 119910 + int
119905and120591
0
120572120601120577
(119884119904) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(204)
showing that 119884 is a local strong solution to (192) with initialcondition 119910 By Itorsquos formula (Theorem 26) we obtain P-almost surely
119883119905and120591
= 120601 (119884119905and120591
)
= ℎ + int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904)))119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(205)
Now let 120585 isin D(119860lowast
) be arbitrary Then we have P-almostsurely
⟨120585 119883119905and120591
⟩
= ⟨120585 ℎ⟩
+ int
119905and120591
0
⟨120585119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 119863120601 (119884119904) 120590
119895
120601120577(119884
119904)⟩119889120573
119895
119904 119905 ge 0
(206)
On the other hand since 119883 is a local weak solution to (176)with initial condition ℎ and lifetime 120591 we have P-almostsurely for all 119905 ge 0 the identity
⟨120585 119883119905and120591
⟩ = ⟨120585 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904)⟩ 119889120573
119895
119904
(207)
Combining (206) and (207) yields up to indistinguishability
119861 + 119872 = 0 (208)
where the processes 119861 and 119872 are defined as
119861119905= int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩) 119889119904
119905 ge 0
119872119905= sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩ 119889120573
119895
119904
119905 ge 0
(209)
The process 119861 + 119872 is a continuous semimartingale withcanonical decomposition (208) Since the canonical decom-position of a continuous semimartingale is unique up toindistinguishability we deduce that 119861 = 119872 = 0 up toindistinguishability Using the Ito isometry (39) we obtain P-almost surely
int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩
+ ⟨120585 120572 (119883119904) minus 119863120601 (119884
showing that119883 is a local strong solution to (44) with lifetime120591
(3)rArr(1) This implication is a direct consequence ofProposition 29
The results from this section are closely related to the exis-tence of finite dimensional realizations that is the existenceof invariantmanifolds for each starting point ℎ
0 andwe point
out the papers [17ndash22] regarding this topic Furthermore wemention thatTheorem 68 has been extended in [23] to SPDEswith jumps
Acknowledgments
The author is grateful to Daniel Gaigall Georg GrafendorferFlorian Modler and Thomas Salfeld for the valuable com-ments and discussions
References
[1] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions CambridgeUniversity Press CambridgeUK 1992
[2] C Prevot and M Rockner A Concise Course on StochasticPartial Differential Equations Springer Berlin Germany 2007
[3] L Gawarecki and V Mandrekar Stochastic Differential Equa-tions in Infinite Dimensions with Applications to StochasticPartial Differential Equations Springer Berlin Germany 2011
[4] S Peszat and J Zabczyk Stochastic Partial Differential Equationswith Levy Noise Cambridge University Press Cambridge UK2007
[5] S Albeverio V Mandrekar and B Rudiger ldquoExistence of mildsolutions for stochastic differential equations and semilinearequations with non-Gaussian Levy noiserdquo Stochastic Processesand Their Applications vol 119 no 3 pp 835ndash863 2009
[6] D Filipovic S Tappe and J Teichmann ldquoJump-diffusions inHilbert spaces existence stability and numericsrdquo Stochasticsvol 82 no 5 pp 475ndash520 2010
[7] C Marinelli C Prevot and M Rockner ldquoRegular dependenceon initial data for stochastic evolution equations with multi-plicative Poisson noiserdquo Journal of Functional Analysis vol 258no 2 pp 616ndash649 2010
[8] S Tappe ldquoSome refinements of existence results for SPDEsdriven by Wiener processes and Poisson random measuresrdquoInternational Journal of Stochastic Analysis vol 2012 Article ID236327 24 pages 2012
[9] T Nakayama ldquoSupport theorem for mild solutions of SDErsquos inHilbert spacesrdquo Journal of Mathematical Sciences the Universityof Tokyo vol 11 no 3 pp 245ndash311 2004
[10] T Nakayama ldquoViability theorem for SPDErsquos including HJMframeworkrdquo Journal of Mathematical Sciences the University ofTokyo vol 11 no 3 pp 313ndash324 2004
[11] W Rudin Functional Analysis McGraw-Hill New York NYUSA 2nd edition 1991
[12] DWerner Funktionalanalysis Springer Berlin Germany 2007[13] C D Aliprantis and K C Border Infinite Dimensional Analysis
Springer Berlin Germany 3rd edition 2006[14] S N Ethier and T G KurtzMarkov Processes Characterization
andConvergenceWiley Series in Probability and Statistics JohnWiley amp Sons New Jersey NJ USA 1986
[15] V Mandrekar and B Rudiger ldquoExistence and uniqueness ofpath wise solutions for stochastic integral equations driven byLevy noise on separable Banach spacesrdquo Stochastics vol 78 no4 pp 189ndash212 2006
[16] D Filipovic Consistency Problems for Heath-Jarrow-MortonInterest Rate Models Springer Berlin Germany 2001
[17] T Bjork and L Svensson ldquoOn the existence of finite-dimensional realizations for nonlinear forward rate modelsrdquoMathematical Finance vol 11 no 2 pp 205ndash243 2001
[18] T Bjork and C Landen ldquoOn the construction of finite dimen-sional realizations for nonlinear forward rate modelsrdquo Financeand Stochastics vol 6 no 3 pp 303ndash331 2002
[19] D Filipovic and J Teichmann ldquoExistence of invariantmanifoldsfor stochastic equations in infinite dimensionrdquo Journal ofFunctional Analysis vol 197 no 2 pp 398ndash432 2003
[20] D Filipovic and J Teichmann ldquoOn the geometry of the termstructure of interest ratesrdquo Proceedings of The Royal Society ofLondon A vol 460 no 2041 pp 129ndash167 2004
[21] S Tappe ldquoAn alternative approach on the existence of affinerealizations for HJM term structure modelsrdquo Proceedings of
International Journal of Stochastic Analysis 25
TheRoyal Society of LondonA vol 466 no 2122 pp 3033ndash30602010
[22] S Tappe ldquoExistence of affine realizations for Levy term struc-ture modelsrdquo Proceedings of The Royal Society of London A vol468 no 2147 pp 3685ndash3704 2012
[23] D Filipovic S Tappe and J Teichmann ldquoInvariant manifoldswith boundary for jump-diffusionsrdquo httparxivorgabs12021076
Since by Proposition 14 the domain D((119860lowast
)2
) is dense in(D(119860
lowast
) sdot D(119860lowast)) we get P-almost surely for all 120577 isin D(119860
lowast
)
the identity
⟨120577 119883119905⟩ = ⟨120577 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904⟩
(70)
Since by Proposition 14 the domainD(119860lowast
) is dense in119867 weobtain P-almost surely
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 (71)
proving that 119883 is a mild solution to (44)
Remark 36 Now the proof of Proposition 32 is an immediateconsequence of Propositions 29 and 35
We have just seen that every weak solution to (44) is also amild solution Under the following regularity condition (72)the converse of this statement holds true as well
Proposition 37 Let 119883 be a mild solution to (44) such that
120572 (119904 119883119904) minus 120572 (119904 119883
119904)⟩ 119889119904
= ⟨120577 int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904⟩ minus int
119905
0
⟨120577 120572 (119904 119883119904)⟩119889119904
(74)
Due to assumption (72) we may use Fubinirsquos theorem forstochastic integrals (see [3 Theorem 28]) which togetherwith Lemma 18 gives us P-almost surely
Due to assumption (80) we may use Fubinirsquos theorem forstochastic integrals (see [3 Theorem 28]) which togetherwith Lemma 18 gives us P-almost surely
int
119905
0
(119878119905minus119904
120590 (119904 119883119904) minus 120590 (119904 119883
This proves that 119883 is also a strong solution to (44)
The following result shows that for norm continuoussemigroups the concepts of strong weak and mild solutionsare equivalent In particular this applies for finite dimen-sional state spaces
Proposition 39 Suppose that the semigroup (119878119905)119905ge0
is normcontinuous Let 119883 be a stochastic process with 119883
0= ℎ
0 Then
the following statements are equivalent
(1) The process 119883 is a strong solution to (44)
(2) The process 119883 is a weak solution to (44)
(3) The process 119883 is a mild solution to (44)
Proof (1)rArr(2) This implication is a consequence ofProposition 29
(2)rArr(3) This implication is a consequence ofProposition 35
(3)rArr(1) By Proposition 20 we have 119860 isin 119871(119867) and 119878119905=
119890119905119860 119905 ge 0 Furthermore the family (119890
119905119860
)119905isinR is a 119862
0-group on
119867 Therefore and since 119883 is a mild solution to (44) we haveP-almost surely
(2) An element 119909 isin 119864 is called a fixed point of Φ if onehas
Φ (119909) = 119909 (109)
The following result is the well-known Banach fixed pointtheorem Its proof can be found for example in [13Theorem348]
Theorem 45 (The Banach fixed point theorem) Let 119864 be acomplete metric space and let Φ 119864 rarr 119864 be a contractionThen the mapping Φ has a unique fixed point
In this text we will use the following slight extension ofthe Banach fixed point theorem
Corollary 46 Let 119864 be a complete metric space and let Φ
119864 rarr 119864 be a mapping such that for some 119899 isin N the mappingΦ119899 is a contraction Then the mapping Φ has a unique fixed
point
Proof According to the Banach fixed point theorem(Theorem 45) the mapping Φ
119899 has a unique fixed point thatis there exists a unique element 119909 isin 119864 such that Φ119899
(119909) = 119909Therefore we have
Φ (119909) = Φ (Φ119899
(119909)) = Φ119899
(Φ (119909)) (110)
showing thatΦ(119909) is a fixedpoint ofΦ119899 SinceΦ119899 has a uniquefixed point we deduce thatΦ(119909) = 119909 showing that119909 is a fixedpoint of Φ
In order to prove uniqueness let 119910 isin 119864 be another fixedpoint ofΦ that is we haveΦ(119910) = 119910 By induction we obtain
showing that 119910 is a fixed point of Φ119899 Since the mapping Φ119899
has exactly one fixed point we obtain 119909 = 119910
An indispensable tool for proving uniqueness of mildsolutions to (44) will be the following version of Gronwallrsquosinequality see for example [14 Theorem 51]
Lemma 47 (Gronwallrsquos inequality) Let 119879 ge 0 be fixed let119891 [0 119879] rarr R
and hence by the Cauchy-Schwarz inequality the Ito isom-etry (39) the growth estimate (9) from Lemma 10 and thelocal Lipschitz conditions (113) we obtain
Using Gronwallrsquos inequality (see Lemma 47) we deduce that119891 equiv 0 Thus by the continuity of the sample paths of 119883 and119884 we obtain
P(⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ) = 1 forall119899 isin N (120)
and hence by the continuity of the probability measureP weconclude that
P(⋂
119905ge0
119883119905and120591
1Γ= 119884
119905and1205911Γ)
= P(⋂
119899isinN
⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ)
= lim119899rarrinfin
P(⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ) = 1
(121)
which completes the proof
The local Lipschitz conditions (113) are in general notsufficient in order to ensure the existence of mild solutions tothe SPDE (44) Now we will prove that the existence of mildsolutions follows from global Lipschitz and linear growthconditions on 120572 and 120590 For this we recall an auxiliary resultwhich extends the Ito isometry (39)
Lemma 49 Let 119879 ge 0 be arbitrary and let 119883 = (119883119905)119905isin[0119879]
for all 119905 ge 0 and all ℎ isin 119867 Then for every F0-measurable
random variable ℎ0
Ω rarr 119867 there exists a (up toindistinguishability) unique mild solution 119883 to (44)
Proof The uniqueness of mild solutions to (44) is a directconsequence of Theorem 48 and hence we may concentrateon the existence proof which we divide into the followingseveral steps
Step 1 First we suppose that the initial condition ℎ0satisfies
E[ℎ02119901
] lt infin for some 119901 gt 1 Let 119879 ge 0 be arbitrary Wedefine the Banach space
1198712119901
119879(119867) = 119871
2119901
(Ω times [0 119879] P119879P otimes 119889119905119867) (129)
and prove that the variation of constants equation
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(130)
has a unique solution in the space 1198712119901
119879(119867) This is done in the
following three steps
Step 11 For 119883 isin 1198712119901
119879(119867) we define the process Φ119883 by
(Φ119883)119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(131)
Then the process Φ119883 is well defined Indeed by the growthestimate (9) the linear growth condition (127) and Holderrsquosinequality we have
Consequently there exists an index 119899 isin N such that Φ119899 is
a contraction and hence according to the extension of theBanach fixed point theorem (see Corollary 46) the mappingΦ has a unique fixed point 119883 isin 119871
2119901
119879(119867) This fixed point 119883 is
a solution to the variation of constants equation (130) Since119879 ge 0 was arbitrary there exists a process 119883 which is a solu-tion of the variation of constants equation
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
119905 ge 0
(146)
Step 14 In order to prove that 119883 is a mild solution to (44)it remains to show that 119883 has a continuous version ByLemma 12 the process
119905 997891997888rarr 119878119905ℎ0 119905 ge 0 (147)
is continuous and by Proposition 41 the process
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 119905 ge 0 (148)
is continuous too Moreover for every 119879 ge 0 we have by thelinear growth condition (128) Holderrsquos inequality and since119883 isin 119871
Thus by Proposition 43 the stochastic convolution 120590 ⋆ 119882
given by
(120590 ⋆ 119882)119905= int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0 (150)
has a continuous version and consequently the process119883hasa continuous version too This continuous version is a mildsolution to (44)
Step 2 Now let ℎ0 Ω rarr 119867 be an arbitraryF
0-measurable
random variable We define the sequence (ℎ119899)119899isinN of F
0-
measurable random variables as
ℎ119899
0= ℎ
01ℎ0le119899
119899 isin N (151)
Let 119899 isin N be arbitrary Then as ℎ119899
0is bounded we have
E[ℎ119899
02119901
] lt infin for all 119901 gt 1 By Step 1 the SPDE
119889119883119899
119905= (119860119883
119899
119905+ 120572 (119905 119883
119899
119905)) 119889119905 + 120590 (119905 119883
119899
119905) 119889119882
119905
119883119899
0= ℎ
119899
0
(152)
18 International Journal of Stochastic Analysis
has amild solution119883119899We define the sequence (Ω
119899)119899isinN sub F
0
as
Ω119899=
1003817100381710038171003817ℎ01003817100381710038171003817 le 119899 119899 isin N (153)
Then we have Ω119899sub Ω
119898for 119899 le 119898 Ω = ⋃
119899isinN Ω119899 and
Ω119899sub ℎ
119899
0= ℎ
119898
0 sub ℎ
119899
0= ℎ
0 forall119899 le 119898 (154)
Thus byTheorem 48 we have (up to indistinguishability)
119883119899
1Ω119899
= 119883119898
1Ω119899
forall119899 le 119898 (155)
Consequently the process
119883 = lim119899rarrinfin
119883119899
1Ω119899
(156)
is a well-defined continuous and adapted process and wehave
119883119899
1Ω119899
= 119883119898
1Ω119899
= 1198831Ω119899
forall119899 le 119898 (157)
Furthermore we obtain P-almost surely
119883119905= lim
119899rarrinfin
119883119899
1199051Ω119899
= lim119899rarrinfin
1Ω119899
(119878119905ℎ119899
0+ int
119905
0
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+int
119905
0
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ119899
0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= lim119899rarrinfin
1Ω119899
(119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= 119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(158)
proving that 119883 is a mild solution to (44)
Remark 51 For the proof of Theorem 50 we have usedCorollary 46 which is a slight extension of the Banach fixedpoint theorem Such an idea has been applied for example in[15]
Remark 52 A recent method for proving existence anduniqueness of mild solutions to the SPDE (44) is the methodof the moving frame presented in [6] see also [8] It allowsto reduce SPDE problems to the study of SDEs in infinitedimension In order to apply this method we need that thesemigroup (119878
119905)119905ge0
is a semigroup of pseudocontractions
We close this section with a consequence about theexistence of weak solutions
Corollary 53 Suppose that conditions (125)ndash(128) are ful-filled Let ℎ
0 Ω rarr 119867 be aF
0-measurable random variable
such that E[ℎ02119901
] lt infin for some 119901 gt 1 Then there exists a(up to indistinguishability) unique weak solution 119883 to (44)
Proof According to Proposition 35 every weak solution119883 to(44) is also a mild solution to (44)Therefore the uniquenessof weak solutions to (44) is a consequence of Theorem 48
It remains to prove the existence of a weak solution to(44) Let 119879 ge 0 be arbitrary By Theorem 50 and its proofthere exists a mild solution119883 isin 119871
2119901
119879(119867) to (44) By the linear
growth condition (128) and Holderrsquos inequality we obtain
showing that condition (72) is fulfilled Thus byProposition 37 the process 119883 is also a weak solution to(44)
8 Invariant Manifolds for WeakSolutions to SPDEs
In this section we deal with invariant manifolds for time-homogeneous SPDEs of the type (44) This topic arises fromthe natural desire to express the solutions of the SPDE (44)which generally live in the infinite dimensional Hilbert space119867 by means of a finite dimensional state process and thusto ensure larger analytical tractability Our goal is to findconditions on the generator 119860 and the coefficients 120572 120590 suchthat for every starting point of a finite dimensional submani-fold the solution process stays on the submanifold
We start with the required preliminaries about finitedimensional submanifolds inHilbert spaces In the sequel let119867 be a separable Hilbert space
Definition 54 Let 119898 119896 isin N be positive integers A subsetM sub 119867 is called an 119898-dimensional 119862119896-submanifold of 119867
International Journal of Stochastic Analysis 19
if for every ℎ isin M there exist an open neighborhood 119880 sub 119867
of ℎ an open set 119881 sub R119898 and a mapping 120601 isin 1198622
(119881119867) suchthat
(1) the mapping 120601 119881 rarr 119880 cap M is a homeomorphism(2) for all 119910 isin 119881 the mapping 119863120601(119910) is injective
The mapping 120601 is called a parametrization ofM around ℎ
In what follows let M be an 119898-dimensional 119862119896-
submanifold of 119867
Lemma 55 Let 120601119894
119881119894
rarr 119880119894cap M 119894 = 1 2 be two
parametrizations with119882= 1198801cap119880
2capM = 0Then themapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (160)
is a 119862119896-diffeomorphism
Proof See [16 Lemma 611]
Corollary 56 Let ℎ isin M be arbitrary and let 120601119894
119881119894
rarr
119880119894capM 119894 = 1 2 be two parametrizations ofM around ℎ Then
one has
1198631206011(119910
1) (R
119898
) = 1198631206012(119910
2) (R
119898
) (161)
where 119910119894= 120601
minus1
119894(ℎ) for 119894 = 1 2
Proof Since1198801and119880
2are open neighborhoods of ℎ we have
119882= 1198801cap 119880
2cap M = 0 Thus by Lemma 55 the mapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (162)
is a 119862119896-diffeomorphism Using the chain rule we obtain
1198631206012(119910
2) (R
119898
)
= 119863 (1206011∘ (120601
minus1
1∘ 120601
2)) (119910
2) (R
119898
)
= 1198631206011(119910
1)119863 (120601
minus1
1∘ 120601
2) (119910
2) (R
119898
)
sub 1198631206011(119910
1) (R
119898
)
(163)
and analogously we prove that 1198631206011(119910
1)(R119898
) sub
1198631206012(119910
2)(R119898
)
Definition 57 Let ℎ isin M be arbitraryThe tangent space ofMto ℎ is the subspace
119879ℎM = 119863120601 (119910) (R
119898
) (164)
where 119910 = 120601minus1
(ℎ) and 120601 119881 rarr 119880 cap M denotes aparametrization ofM around ℎ
Remark 58 Note that according to Corollary 56 theDefinition 57 of the tangent space 119879
ℎM does not depend on
the choice of the parametrization 120601 119881 rarr 119880 cap M
Proposition 59 Let ℎ isin M be arbitrary and let 120601 119881 rarr
119880 cap M be a parametrization ofM around ℎ Then there existan open set 119881
0sub 119881 an open neighborhood 119880
0sub 119880 of ℎ and
a mapping 120601 isin 119862119896
119887(R119898
119867) with 120601|1198810
= 120601|1198810
such that 120601|1198810
1198810
rarr 1198800cap M is a parametrization ofM around ℎ too
Proof See [16 Remark 611]
Remark 60 By Proposition 59 we may assume that anyparametrization 120601 119881 rarr 119880 cap M has an extension 120601 isin
119862119896
119887(R119898
119867)
Proposition 61 Let 119863 sub 119867 be a dense subset For every ℎ0isin
M there exist 1205771 120577
119898isin 119863 and a parametrization 120601 119881 rarr
119880 cap M around ℎ0such that
120601 (⟨120577 ℎ⟩) = ℎ forallℎ isin 119880 cap M (165)
where one uses the notation ⟨120577 ℎ⟩ = (⟨1205771 ℎ⟩ ⟨120577
119898 ℎ⟩) isin
R119898
Proof See [16 Proposition 612]
Proposition 62 Let 120601 119881 rarr 119880 cap M be a parametrizationas in Proposition 61 Then the following statements are true
(1) The elements 1205771 120577
119898are linearly independent in 119867
(2) For every ℎ isin 119880 cap M one has the direct sum decom-position
Therefore by Proposition 64 for every 1199100isin R119898 there exists
a (up to indistinguishability) unique weak solution to (192)which according to Proposition 39 is also a strong solutionto (192) The uniqueness of strong solutions to (192) is a con-sequence of Proposition 39 andTheorem 48
Now we are ready to formulate and prove ourmain resultof this section
Theorem 68 The following statements are equivalent
(1) The submanifoldM is locally invariant for (176)(2) One has
M sub D (119860) (195)
120590119895
(ℎ) isin 119879ℎM forallℎ isin M 119886119897119897 119895 isin N (196)
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) isin 119879ℎM forallℎ isin M (197)
(3) The operator 119860 is continuous onM and for each ℎ0isin
M there exists a local strong solution 119883 to (176) withsome lifetime 120591 gt 0 such that
119883119905and120591
isin M forall119905 ge 0 P-119886119897119898119900119904119905 119904119906119903119890119897119910 (198)
Proof (1)rArr(2) Let ℎ isin M be arbitrary By Proposition 61and Remark 60 there exist elements 120577
1 120577
119898isin D(119860
lowast
) anda parametrization 120601 119881 rarr 119880 cap M around ℎ such that theinverse 120601
minus1
119880 cap M rarr 119881 is given by 120601minus1
= ⟨120577 ∙⟩ and 120601
has an extension 120601 isin 1198622
119887(R119898
119867) Since the submanifoldM islocally invariant for (176) there exists a local weak solution119883
to (176) with initial condition ℎ and some lifetime 984858 gt 0 suchthat
119883119905and984858
isin M forall119905 ge 0 P-almost surely (199)
Since119880 is an open neighborhood of ℎ there exists 120598 gt 0 suchthat 119861
120598(ℎ) sub 119880 where 119861
120598(ℎ) denotes the open ball
119861120598(ℎ) = 119892 isin 119867
1003817100381710038171003817119892 minus ℎ1003817100381710038171003817 lt 120598 (200)
We define the stopping time
120591 = 984858 and inf 119905 ge 0 119883119905notin 119861
120598(ℎ) (201)
Since the process119883 has continuous sample paths and satisfies1198830= ℎ we have 120591 gt 0 and P-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (202)
22 International Journal of Stochastic Analysis
Defining the R119898-valued process 119884 = ⟨120577 119883⟩ we have P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (203)
Moreover since 119883 is a weak solution to (176) with initialcondition ℎ setting 119910 = ⟨120577 ℎ⟩ isin 119881 we have P-almost surely
119884119905and120591
= ⟨120577 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120577 120590119895
(119883119904)⟩ 119889120573
119895
119904
= ⟨120577 ℎ⟩ + int
119905and120591
0
120572120601120577
(⟨120577 119883119904⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(⟨120577 119883
119904⟩) 119889120573
119895
119904
= 119910 + int
119905and120591
0
120572120601120577
(119884119904) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(204)
showing that 119884 is a local strong solution to (192) with initialcondition 119910 By Itorsquos formula (Theorem 26) we obtain P-almost surely
119883119905and120591
= 120601 (119884119905and120591
)
= ℎ + int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904)))119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(205)
Now let 120585 isin D(119860lowast
) be arbitrary Then we have P-almostsurely
⟨120585 119883119905and120591
⟩
= ⟨120585 ℎ⟩
+ int
119905and120591
0
⟨120585119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 119863120601 (119884119904) 120590
119895
120601120577(119884
119904)⟩119889120573
119895
119904 119905 ge 0
(206)
On the other hand since 119883 is a local weak solution to (176)with initial condition ℎ and lifetime 120591 we have P-almostsurely for all 119905 ge 0 the identity
⟨120585 119883119905and120591
⟩ = ⟨120585 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904)⟩ 119889120573
119895
119904
(207)
Combining (206) and (207) yields up to indistinguishability
119861 + 119872 = 0 (208)
where the processes 119861 and 119872 are defined as
119861119905= int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩) 119889119904
119905 ge 0
119872119905= sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩ 119889120573
119895
119904
119905 ge 0
(209)
The process 119861 + 119872 is a continuous semimartingale withcanonical decomposition (208) Since the canonical decom-position of a continuous semimartingale is unique up toindistinguishability we deduce that 119861 = 119872 = 0 up toindistinguishability Using the Ito isometry (39) we obtain P-almost surely
int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩
+ ⟨120585 120572 (119883119904) minus 119863120601 (119884
showing that119883 is a local strong solution to (44) with lifetime120591
(3)rArr(1) This implication is a direct consequence ofProposition 29
The results from this section are closely related to the exis-tence of finite dimensional realizations that is the existenceof invariantmanifolds for each starting point ℎ
0 andwe point
out the papers [17ndash22] regarding this topic Furthermore wemention thatTheorem 68 has been extended in [23] to SPDEswith jumps
Acknowledgments
The author is grateful to Daniel Gaigall Georg GrafendorferFlorian Modler and Thomas Salfeld for the valuable com-ments and discussions
References
[1] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions CambridgeUniversity Press CambridgeUK 1992
[2] C Prevot and M Rockner A Concise Course on StochasticPartial Differential Equations Springer Berlin Germany 2007
[3] L Gawarecki and V Mandrekar Stochastic Differential Equa-tions in Infinite Dimensions with Applications to StochasticPartial Differential Equations Springer Berlin Germany 2011
[4] S Peszat and J Zabczyk Stochastic Partial Differential Equationswith Levy Noise Cambridge University Press Cambridge UK2007
[5] S Albeverio V Mandrekar and B Rudiger ldquoExistence of mildsolutions for stochastic differential equations and semilinearequations with non-Gaussian Levy noiserdquo Stochastic Processesand Their Applications vol 119 no 3 pp 835ndash863 2009
[6] D Filipovic S Tappe and J Teichmann ldquoJump-diffusions inHilbert spaces existence stability and numericsrdquo Stochasticsvol 82 no 5 pp 475ndash520 2010
[7] C Marinelli C Prevot and M Rockner ldquoRegular dependenceon initial data for stochastic evolution equations with multi-plicative Poisson noiserdquo Journal of Functional Analysis vol 258no 2 pp 616ndash649 2010
[8] S Tappe ldquoSome refinements of existence results for SPDEsdriven by Wiener processes and Poisson random measuresrdquoInternational Journal of Stochastic Analysis vol 2012 Article ID236327 24 pages 2012
[9] T Nakayama ldquoSupport theorem for mild solutions of SDErsquos inHilbert spacesrdquo Journal of Mathematical Sciences the Universityof Tokyo vol 11 no 3 pp 245ndash311 2004
[10] T Nakayama ldquoViability theorem for SPDErsquos including HJMframeworkrdquo Journal of Mathematical Sciences the University ofTokyo vol 11 no 3 pp 313ndash324 2004
[11] W Rudin Functional Analysis McGraw-Hill New York NYUSA 2nd edition 1991
[12] DWerner Funktionalanalysis Springer Berlin Germany 2007[13] C D Aliprantis and K C Border Infinite Dimensional Analysis
Springer Berlin Germany 3rd edition 2006[14] S N Ethier and T G KurtzMarkov Processes Characterization
andConvergenceWiley Series in Probability and Statistics JohnWiley amp Sons New Jersey NJ USA 1986
[15] V Mandrekar and B Rudiger ldquoExistence and uniqueness ofpath wise solutions for stochastic integral equations driven byLevy noise on separable Banach spacesrdquo Stochastics vol 78 no4 pp 189ndash212 2006
[16] D Filipovic Consistency Problems for Heath-Jarrow-MortonInterest Rate Models Springer Berlin Germany 2001
[17] T Bjork and L Svensson ldquoOn the existence of finite-dimensional realizations for nonlinear forward rate modelsrdquoMathematical Finance vol 11 no 2 pp 205ndash243 2001
[18] T Bjork and C Landen ldquoOn the construction of finite dimen-sional realizations for nonlinear forward rate modelsrdquo Financeand Stochastics vol 6 no 3 pp 303ndash331 2002
[19] D Filipovic and J Teichmann ldquoExistence of invariantmanifoldsfor stochastic equations in infinite dimensionrdquo Journal ofFunctional Analysis vol 197 no 2 pp 398ndash432 2003
[20] D Filipovic and J Teichmann ldquoOn the geometry of the termstructure of interest ratesrdquo Proceedings of The Royal Society ofLondon A vol 460 no 2041 pp 129ndash167 2004
[21] S Tappe ldquoAn alternative approach on the existence of affinerealizations for HJM term structure modelsrdquo Proceedings of
International Journal of Stochastic Analysis 25
TheRoyal Society of LondonA vol 466 no 2122 pp 3033ndash30602010
[22] S Tappe ldquoExistence of affine realizations for Levy term struc-ture modelsrdquo Proceedings of The Royal Society of London A vol468 no 2147 pp 3685ndash3704 2012
[23] D Filipovic S Tappe and J Teichmann ldquoInvariant manifoldswith boundary for jump-diffusionsrdquo httparxivorgabs12021076
Due to assumption (80) we may use Fubinirsquos theorem forstochastic integrals (see [3 Theorem 28]) which togetherwith Lemma 18 gives us P-almost surely
int
119905
0
(119878119905minus119904
120590 (119904 119883119904) minus 120590 (119904 119883
This proves that 119883 is also a strong solution to (44)
The following result shows that for norm continuoussemigroups the concepts of strong weak and mild solutionsare equivalent In particular this applies for finite dimen-sional state spaces
Proposition 39 Suppose that the semigroup (119878119905)119905ge0
is normcontinuous Let 119883 be a stochastic process with 119883
0= ℎ
0 Then
the following statements are equivalent
(1) The process 119883 is a strong solution to (44)
(2) The process 119883 is a weak solution to (44)
(3) The process 119883 is a mild solution to (44)
Proof (1)rArr(2) This implication is a consequence ofProposition 29
(2)rArr(3) This implication is a consequence ofProposition 35
(3)rArr(1) By Proposition 20 we have 119860 isin 119871(119867) and 119878119905=
119890119905119860 119905 ge 0 Furthermore the family (119890
119905119860
)119905isinR is a 119862
0-group on
119867 Therefore and since 119883 is a mild solution to (44) we haveP-almost surely
(2) An element 119909 isin 119864 is called a fixed point of Φ if onehas
Φ (119909) = 119909 (109)
The following result is the well-known Banach fixed pointtheorem Its proof can be found for example in [13Theorem348]
Theorem 45 (The Banach fixed point theorem) Let 119864 be acomplete metric space and let Φ 119864 rarr 119864 be a contractionThen the mapping Φ has a unique fixed point
In this text we will use the following slight extension ofthe Banach fixed point theorem
Corollary 46 Let 119864 be a complete metric space and let Φ
119864 rarr 119864 be a mapping such that for some 119899 isin N the mappingΦ119899 is a contraction Then the mapping Φ has a unique fixed
point
Proof According to the Banach fixed point theorem(Theorem 45) the mapping Φ
119899 has a unique fixed point thatis there exists a unique element 119909 isin 119864 such that Φ119899
(119909) = 119909Therefore we have
Φ (119909) = Φ (Φ119899
(119909)) = Φ119899
(Φ (119909)) (110)
showing thatΦ(119909) is a fixedpoint ofΦ119899 SinceΦ119899 has a uniquefixed point we deduce thatΦ(119909) = 119909 showing that119909 is a fixedpoint of Φ
In order to prove uniqueness let 119910 isin 119864 be another fixedpoint ofΦ that is we haveΦ(119910) = 119910 By induction we obtain
showing that 119910 is a fixed point of Φ119899 Since the mapping Φ119899
has exactly one fixed point we obtain 119909 = 119910
An indispensable tool for proving uniqueness of mildsolutions to (44) will be the following version of Gronwallrsquosinequality see for example [14 Theorem 51]
Lemma 47 (Gronwallrsquos inequality) Let 119879 ge 0 be fixed let119891 [0 119879] rarr R
and hence by the Cauchy-Schwarz inequality the Ito isom-etry (39) the growth estimate (9) from Lemma 10 and thelocal Lipschitz conditions (113) we obtain
Using Gronwallrsquos inequality (see Lemma 47) we deduce that119891 equiv 0 Thus by the continuity of the sample paths of 119883 and119884 we obtain
P(⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ) = 1 forall119899 isin N (120)
and hence by the continuity of the probability measureP weconclude that
P(⋂
119905ge0
119883119905and120591
1Γ= 119884
119905and1205911Γ)
= P(⋂
119899isinN
⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ)
= lim119899rarrinfin
P(⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ) = 1
(121)
which completes the proof
The local Lipschitz conditions (113) are in general notsufficient in order to ensure the existence of mild solutions tothe SPDE (44) Now we will prove that the existence of mildsolutions follows from global Lipschitz and linear growthconditions on 120572 and 120590 For this we recall an auxiliary resultwhich extends the Ito isometry (39)
Lemma 49 Let 119879 ge 0 be arbitrary and let 119883 = (119883119905)119905isin[0119879]
for all 119905 ge 0 and all ℎ isin 119867 Then for every F0-measurable
random variable ℎ0
Ω rarr 119867 there exists a (up toindistinguishability) unique mild solution 119883 to (44)
Proof The uniqueness of mild solutions to (44) is a directconsequence of Theorem 48 and hence we may concentrateon the existence proof which we divide into the followingseveral steps
Step 1 First we suppose that the initial condition ℎ0satisfies
E[ℎ02119901
] lt infin for some 119901 gt 1 Let 119879 ge 0 be arbitrary Wedefine the Banach space
1198712119901
119879(119867) = 119871
2119901
(Ω times [0 119879] P119879P otimes 119889119905119867) (129)
and prove that the variation of constants equation
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(130)
has a unique solution in the space 1198712119901
119879(119867) This is done in the
following three steps
Step 11 For 119883 isin 1198712119901
119879(119867) we define the process Φ119883 by
(Φ119883)119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(131)
Then the process Φ119883 is well defined Indeed by the growthestimate (9) the linear growth condition (127) and Holderrsquosinequality we have
Consequently there exists an index 119899 isin N such that Φ119899 is
a contraction and hence according to the extension of theBanach fixed point theorem (see Corollary 46) the mappingΦ has a unique fixed point 119883 isin 119871
2119901
119879(119867) This fixed point 119883 is
a solution to the variation of constants equation (130) Since119879 ge 0 was arbitrary there exists a process 119883 which is a solu-tion of the variation of constants equation
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
119905 ge 0
(146)
Step 14 In order to prove that 119883 is a mild solution to (44)it remains to show that 119883 has a continuous version ByLemma 12 the process
119905 997891997888rarr 119878119905ℎ0 119905 ge 0 (147)
is continuous and by Proposition 41 the process
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 119905 ge 0 (148)
is continuous too Moreover for every 119879 ge 0 we have by thelinear growth condition (128) Holderrsquos inequality and since119883 isin 119871
Thus by Proposition 43 the stochastic convolution 120590 ⋆ 119882
given by
(120590 ⋆ 119882)119905= int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0 (150)
has a continuous version and consequently the process119883hasa continuous version too This continuous version is a mildsolution to (44)
Step 2 Now let ℎ0 Ω rarr 119867 be an arbitraryF
0-measurable
random variable We define the sequence (ℎ119899)119899isinN of F
0-
measurable random variables as
ℎ119899
0= ℎ
01ℎ0le119899
119899 isin N (151)
Let 119899 isin N be arbitrary Then as ℎ119899
0is bounded we have
E[ℎ119899
02119901
] lt infin for all 119901 gt 1 By Step 1 the SPDE
119889119883119899
119905= (119860119883
119899
119905+ 120572 (119905 119883
119899
119905)) 119889119905 + 120590 (119905 119883
119899
119905) 119889119882
119905
119883119899
0= ℎ
119899
0
(152)
18 International Journal of Stochastic Analysis
has amild solution119883119899We define the sequence (Ω
119899)119899isinN sub F
0
as
Ω119899=
1003817100381710038171003817ℎ01003817100381710038171003817 le 119899 119899 isin N (153)
Then we have Ω119899sub Ω
119898for 119899 le 119898 Ω = ⋃
119899isinN Ω119899 and
Ω119899sub ℎ
119899
0= ℎ
119898
0 sub ℎ
119899
0= ℎ
0 forall119899 le 119898 (154)
Thus byTheorem 48 we have (up to indistinguishability)
119883119899
1Ω119899
= 119883119898
1Ω119899
forall119899 le 119898 (155)
Consequently the process
119883 = lim119899rarrinfin
119883119899
1Ω119899
(156)
is a well-defined continuous and adapted process and wehave
119883119899
1Ω119899
= 119883119898
1Ω119899
= 1198831Ω119899
forall119899 le 119898 (157)
Furthermore we obtain P-almost surely
119883119905= lim
119899rarrinfin
119883119899
1199051Ω119899
= lim119899rarrinfin
1Ω119899
(119878119905ℎ119899
0+ int
119905
0
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+int
119905
0
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ119899
0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= lim119899rarrinfin
1Ω119899
(119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= 119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(158)
proving that 119883 is a mild solution to (44)
Remark 51 For the proof of Theorem 50 we have usedCorollary 46 which is a slight extension of the Banach fixedpoint theorem Such an idea has been applied for example in[15]
Remark 52 A recent method for proving existence anduniqueness of mild solutions to the SPDE (44) is the methodof the moving frame presented in [6] see also [8] It allowsto reduce SPDE problems to the study of SDEs in infinitedimension In order to apply this method we need that thesemigroup (119878
119905)119905ge0
is a semigroup of pseudocontractions
We close this section with a consequence about theexistence of weak solutions
Corollary 53 Suppose that conditions (125)ndash(128) are ful-filled Let ℎ
0 Ω rarr 119867 be aF
0-measurable random variable
such that E[ℎ02119901
] lt infin for some 119901 gt 1 Then there exists a(up to indistinguishability) unique weak solution 119883 to (44)
Proof According to Proposition 35 every weak solution119883 to(44) is also a mild solution to (44)Therefore the uniquenessof weak solutions to (44) is a consequence of Theorem 48
It remains to prove the existence of a weak solution to(44) Let 119879 ge 0 be arbitrary By Theorem 50 and its proofthere exists a mild solution119883 isin 119871
2119901
119879(119867) to (44) By the linear
growth condition (128) and Holderrsquos inequality we obtain
showing that condition (72) is fulfilled Thus byProposition 37 the process 119883 is also a weak solution to(44)
8 Invariant Manifolds for WeakSolutions to SPDEs
In this section we deal with invariant manifolds for time-homogeneous SPDEs of the type (44) This topic arises fromthe natural desire to express the solutions of the SPDE (44)which generally live in the infinite dimensional Hilbert space119867 by means of a finite dimensional state process and thusto ensure larger analytical tractability Our goal is to findconditions on the generator 119860 and the coefficients 120572 120590 suchthat for every starting point of a finite dimensional submani-fold the solution process stays on the submanifold
We start with the required preliminaries about finitedimensional submanifolds inHilbert spaces In the sequel let119867 be a separable Hilbert space
Definition 54 Let 119898 119896 isin N be positive integers A subsetM sub 119867 is called an 119898-dimensional 119862119896-submanifold of 119867
International Journal of Stochastic Analysis 19
if for every ℎ isin M there exist an open neighborhood 119880 sub 119867
of ℎ an open set 119881 sub R119898 and a mapping 120601 isin 1198622
(119881119867) suchthat
(1) the mapping 120601 119881 rarr 119880 cap M is a homeomorphism(2) for all 119910 isin 119881 the mapping 119863120601(119910) is injective
The mapping 120601 is called a parametrization ofM around ℎ
In what follows let M be an 119898-dimensional 119862119896-
submanifold of 119867
Lemma 55 Let 120601119894
119881119894
rarr 119880119894cap M 119894 = 1 2 be two
parametrizations with119882= 1198801cap119880
2capM = 0Then themapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (160)
is a 119862119896-diffeomorphism
Proof See [16 Lemma 611]
Corollary 56 Let ℎ isin M be arbitrary and let 120601119894
119881119894
rarr
119880119894capM 119894 = 1 2 be two parametrizations ofM around ℎ Then
one has
1198631206011(119910
1) (R
119898
) = 1198631206012(119910
2) (R
119898
) (161)
where 119910119894= 120601
minus1
119894(ℎ) for 119894 = 1 2
Proof Since1198801and119880
2are open neighborhoods of ℎ we have
119882= 1198801cap 119880
2cap M = 0 Thus by Lemma 55 the mapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (162)
is a 119862119896-diffeomorphism Using the chain rule we obtain
1198631206012(119910
2) (R
119898
)
= 119863 (1206011∘ (120601
minus1
1∘ 120601
2)) (119910
2) (R
119898
)
= 1198631206011(119910
1)119863 (120601
minus1
1∘ 120601
2) (119910
2) (R
119898
)
sub 1198631206011(119910
1) (R
119898
)
(163)
and analogously we prove that 1198631206011(119910
1)(R119898
) sub
1198631206012(119910
2)(R119898
)
Definition 57 Let ℎ isin M be arbitraryThe tangent space ofMto ℎ is the subspace
119879ℎM = 119863120601 (119910) (R
119898
) (164)
where 119910 = 120601minus1
(ℎ) and 120601 119881 rarr 119880 cap M denotes aparametrization ofM around ℎ
Remark 58 Note that according to Corollary 56 theDefinition 57 of the tangent space 119879
ℎM does not depend on
the choice of the parametrization 120601 119881 rarr 119880 cap M
Proposition 59 Let ℎ isin M be arbitrary and let 120601 119881 rarr
119880 cap M be a parametrization ofM around ℎ Then there existan open set 119881
0sub 119881 an open neighborhood 119880
0sub 119880 of ℎ and
a mapping 120601 isin 119862119896
119887(R119898
119867) with 120601|1198810
= 120601|1198810
such that 120601|1198810
1198810
rarr 1198800cap M is a parametrization ofM around ℎ too
Proof See [16 Remark 611]
Remark 60 By Proposition 59 we may assume that anyparametrization 120601 119881 rarr 119880 cap M has an extension 120601 isin
119862119896
119887(R119898
119867)
Proposition 61 Let 119863 sub 119867 be a dense subset For every ℎ0isin
M there exist 1205771 120577
119898isin 119863 and a parametrization 120601 119881 rarr
119880 cap M around ℎ0such that
120601 (⟨120577 ℎ⟩) = ℎ forallℎ isin 119880 cap M (165)
where one uses the notation ⟨120577 ℎ⟩ = (⟨1205771 ℎ⟩ ⟨120577
119898 ℎ⟩) isin
R119898
Proof See [16 Proposition 612]
Proposition 62 Let 120601 119881 rarr 119880 cap M be a parametrizationas in Proposition 61 Then the following statements are true
(1) The elements 1205771 120577
119898are linearly independent in 119867
(2) For every ℎ isin 119880 cap M one has the direct sum decom-position
Therefore by Proposition 64 for every 1199100isin R119898 there exists
a (up to indistinguishability) unique weak solution to (192)which according to Proposition 39 is also a strong solutionto (192) The uniqueness of strong solutions to (192) is a con-sequence of Proposition 39 andTheorem 48
Now we are ready to formulate and prove ourmain resultof this section
Theorem 68 The following statements are equivalent
(1) The submanifoldM is locally invariant for (176)(2) One has
M sub D (119860) (195)
120590119895
(ℎ) isin 119879ℎM forallℎ isin M 119886119897119897 119895 isin N (196)
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) isin 119879ℎM forallℎ isin M (197)
(3) The operator 119860 is continuous onM and for each ℎ0isin
M there exists a local strong solution 119883 to (176) withsome lifetime 120591 gt 0 such that
119883119905and120591
isin M forall119905 ge 0 P-119886119897119898119900119904119905 119904119906119903119890119897119910 (198)
Proof (1)rArr(2) Let ℎ isin M be arbitrary By Proposition 61and Remark 60 there exist elements 120577
1 120577
119898isin D(119860
lowast
) anda parametrization 120601 119881 rarr 119880 cap M around ℎ such that theinverse 120601
minus1
119880 cap M rarr 119881 is given by 120601minus1
= ⟨120577 ∙⟩ and 120601
has an extension 120601 isin 1198622
119887(R119898
119867) Since the submanifoldM islocally invariant for (176) there exists a local weak solution119883
to (176) with initial condition ℎ and some lifetime 984858 gt 0 suchthat
119883119905and984858
isin M forall119905 ge 0 P-almost surely (199)
Since119880 is an open neighborhood of ℎ there exists 120598 gt 0 suchthat 119861
120598(ℎ) sub 119880 where 119861
120598(ℎ) denotes the open ball
119861120598(ℎ) = 119892 isin 119867
1003817100381710038171003817119892 minus ℎ1003817100381710038171003817 lt 120598 (200)
We define the stopping time
120591 = 984858 and inf 119905 ge 0 119883119905notin 119861
120598(ℎ) (201)
Since the process119883 has continuous sample paths and satisfies1198830= ℎ we have 120591 gt 0 and P-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (202)
22 International Journal of Stochastic Analysis
Defining the R119898-valued process 119884 = ⟨120577 119883⟩ we have P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (203)
Moreover since 119883 is a weak solution to (176) with initialcondition ℎ setting 119910 = ⟨120577 ℎ⟩ isin 119881 we have P-almost surely
119884119905and120591
= ⟨120577 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120577 120590119895
(119883119904)⟩ 119889120573
119895
119904
= ⟨120577 ℎ⟩ + int
119905and120591
0
120572120601120577
(⟨120577 119883119904⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(⟨120577 119883
119904⟩) 119889120573
119895
119904
= 119910 + int
119905and120591
0
120572120601120577
(119884119904) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(204)
showing that 119884 is a local strong solution to (192) with initialcondition 119910 By Itorsquos formula (Theorem 26) we obtain P-almost surely
119883119905and120591
= 120601 (119884119905and120591
)
= ℎ + int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904)))119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(205)
Now let 120585 isin D(119860lowast
) be arbitrary Then we have P-almostsurely
⟨120585 119883119905and120591
⟩
= ⟨120585 ℎ⟩
+ int
119905and120591
0
⟨120585119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 119863120601 (119884119904) 120590
119895
120601120577(119884
119904)⟩119889120573
119895
119904 119905 ge 0
(206)
On the other hand since 119883 is a local weak solution to (176)with initial condition ℎ and lifetime 120591 we have P-almostsurely for all 119905 ge 0 the identity
⟨120585 119883119905and120591
⟩ = ⟨120585 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904)⟩ 119889120573
119895
119904
(207)
Combining (206) and (207) yields up to indistinguishability
119861 + 119872 = 0 (208)
where the processes 119861 and 119872 are defined as
119861119905= int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩) 119889119904
119905 ge 0
119872119905= sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩ 119889120573
119895
119904
119905 ge 0
(209)
The process 119861 + 119872 is a continuous semimartingale withcanonical decomposition (208) Since the canonical decom-position of a continuous semimartingale is unique up toindistinguishability we deduce that 119861 = 119872 = 0 up toindistinguishability Using the Ito isometry (39) we obtain P-almost surely
int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩
+ ⟨120585 120572 (119883119904) minus 119863120601 (119884
showing that119883 is a local strong solution to (44) with lifetime120591
(3)rArr(1) This implication is a direct consequence ofProposition 29
The results from this section are closely related to the exis-tence of finite dimensional realizations that is the existenceof invariantmanifolds for each starting point ℎ
0 andwe point
out the papers [17ndash22] regarding this topic Furthermore wemention thatTheorem 68 has been extended in [23] to SPDEswith jumps
Acknowledgments
The author is grateful to Daniel Gaigall Georg GrafendorferFlorian Modler and Thomas Salfeld for the valuable com-ments and discussions
References
[1] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions CambridgeUniversity Press CambridgeUK 1992
[2] C Prevot and M Rockner A Concise Course on StochasticPartial Differential Equations Springer Berlin Germany 2007
[3] L Gawarecki and V Mandrekar Stochastic Differential Equa-tions in Infinite Dimensions with Applications to StochasticPartial Differential Equations Springer Berlin Germany 2011
[4] S Peszat and J Zabczyk Stochastic Partial Differential Equationswith Levy Noise Cambridge University Press Cambridge UK2007
[5] S Albeverio V Mandrekar and B Rudiger ldquoExistence of mildsolutions for stochastic differential equations and semilinearequations with non-Gaussian Levy noiserdquo Stochastic Processesand Their Applications vol 119 no 3 pp 835ndash863 2009
[6] D Filipovic S Tappe and J Teichmann ldquoJump-diffusions inHilbert spaces existence stability and numericsrdquo Stochasticsvol 82 no 5 pp 475ndash520 2010
[7] C Marinelli C Prevot and M Rockner ldquoRegular dependenceon initial data for stochastic evolution equations with multi-plicative Poisson noiserdquo Journal of Functional Analysis vol 258no 2 pp 616ndash649 2010
[8] S Tappe ldquoSome refinements of existence results for SPDEsdriven by Wiener processes and Poisson random measuresrdquoInternational Journal of Stochastic Analysis vol 2012 Article ID236327 24 pages 2012
[9] T Nakayama ldquoSupport theorem for mild solutions of SDErsquos inHilbert spacesrdquo Journal of Mathematical Sciences the Universityof Tokyo vol 11 no 3 pp 245ndash311 2004
[10] T Nakayama ldquoViability theorem for SPDErsquos including HJMframeworkrdquo Journal of Mathematical Sciences the University ofTokyo vol 11 no 3 pp 313ndash324 2004
[11] W Rudin Functional Analysis McGraw-Hill New York NYUSA 2nd edition 1991
[12] DWerner Funktionalanalysis Springer Berlin Germany 2007[13] C D Aliprantis and K C Border Infinite Dimensional Analysis
Springer Berlin Germany 3rd edition 2006[14] S N Ethier and T G KurtzMarkov Processes Characterization
andConvergenceWiley Series in Probability and Statistics JohnWiley amp Sons New Jersey NJ USA 1986
[15] V Mandrekar and B Rudiger ldquoExistence and uniqueness ofpath wise solutions for stochastic integral equations driven byLevy noise on separable Banach spacesrdquo Stochastics vol 78 no4 pp 189ndash212 2006
[16] D Filipovic Consistency Problems for Heath-Jarrow-MortonInterest Rate Models Springer Berlin Germany 2001
[17] T Bjork and L Svensson ldquoOn the existence of finite-dimensional realizations for nonlinear forward rate modelsrdquoMathematical Finance vol 11 no 2 pp 205ndash243 2001
[18] T Bjork and C Landen ldquoOn the construction of finite dimen-sional realizations for nonlinear forward rate modelsrdquo Financeand Stochastics vol 6 no 3 pp 303ndash331 2002
[19] D Filipovic and J Teichmann ldquoExistence of invariantmanifoldsfor stochastic equations in infinite dimensionrdquo Journal ofFunctional Analysis vol 197 no 2 pp 398ndash432 2003
[20] D Filipovic and J Teichmann ldquoOn the geometry of the termstructure of interest ratesrdquo Proceedings of The Royal Society ofLondon A vol 460 no 2041 pp 129ndash167 2004
[21] S Tappe ldquoAn alternative approach on the existence of affinerealizations for HJM term structure modelsrdquo Proceedings of
International Journal of Stochastic Analysis 25
TheRoyal Society of LondonA vol 466 no 2122 pp 3033ndash30602010
[22] S Tappe ldquoExistence of affine realizations for Levy term struc-ture modelsrdquo Proceedings of The Royal Society of London A vol468 no 2147 pp 3685ndash3704 2012
[23] D Filipovic S Tappe and J Teichmann ldquoInvariant manifoldswith boundary for jump-diffusionsrdquo httparxivorgabs12021076
This proves that 119883 is also a strong solution to (44)
The following result shows that for norm continuoussemigroups the concepts of strong weak and mild solutionsare equivalent In particular this applies for finite dimen-sional state spaces
Proposition 39 Suppose that the semigroup (119878119905)119905ge0
is normcontinuous Let 119883 be a stochastic process with 119883
0= ℎ
0 Then
the following statements are equivalent
(1) The process 119883 is a strong solution to (44)
(2) The process 119883 is a weak solution to (44)
(3) The process 119883 is a mild solution to (44)
Proof (1)rArr(2) This implication is a consequence ofProposition 29
(2)rArr(3) This implication is a consequence ofProposition 35
(3)rArr(1) By Proposition 20 we have 119860 isin 119871(119867) and 119878119905=
119890119905119860 119905 ge 0 Furthermore the family (119890
119905119860
)119905isinR is a 119862
0-group on
119867 Therefore and since 119883 is a mild solution to (44) we haveP-almost surely
(2) An element 119909 isin 119864 is called a fixed point of Φ if onehas
Φ (119909) = 119909 (109)
The following result is the well-known Banach fixed pointtheorem Its proof can be found for example in [13Theorem348]
Theorem 45 (The Banach fixed point theorem) Let 119864 be acomplete metric space and let Φ 119864 rarr 119864 be a contractionThen the mapping Φ has a unique fixed point
In this text we will use the following slight extension ofthe Banach fixed point theorem
Corollary 46 Let 119864 be a complete metric space and let Φ
119864 rarr 119864 be a mapping such that for some 119899 isin N the mappingΦ119899 is a contraction Then the mapping Φ has a unique fixed
point
Proof According to the Banach fixed point theorem(Theorem 45) the mapping Φ
119899 has a unique fixed point thatis there exists a unique element 119909 isin 119864 such that Φ119899
(119909) = 119909Therefore we have
Φ (119909) = Φ (Φ119899
(119909)) = Φ119899
(Φ (119909)) (110)
showing thatΦ(119909) is a fixedpoint ofΦ119899 SinceΦ119899 has a uniquefixed point we deduce thatΦ(119909) = 119909 showing that119909 is a fixedpoint of Φ
In order to prove uniqueness let 119910 isin 119864 be another fixedpoint ofΦ that is we haveΦ(119910) = 119910 By induction we obtain
showing that 119910 is a fixed point of Φ119899 Since the mapping Φ119899
has exactly one fixed point we obtain 119909 = 119910
An indispensable tool for proving uniqueness of mildsolutions to (44) will be the following version of Gronwallrsquosinequality see for example [14 Theorem 51]
Lemma 47 (Gronwallrsquos inequality) Let 119879 ge 0 be fixed let119891 [0 119879] rarr R
and hence by the Cauchy-Schwarz inequality the Ito isom-etry (39) the growth estimate (9) from Lemma 10 and thelocal Lipschitz conditions (113) we obtain
Using Gronwallrsquos inequality (see Lemma 47) we deduce that119891 equiv 0 Thus by the continuity of the sample paths of 119883 and119884 we obtain
P(⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ) = 1 forall119899 isin N (120)
and hence by the continuity of the probability measureP weconclude that
P(⋂
119905ge0
119883119905and120591
1Γ= 119884
119905and1205911Γ)
= P(⋂
119899isinN
⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ)
= lim119899rarrinfin
P(⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ) = 1
(121)
which completes the proof
The local Lipschitz conditions (113) are in general notsufficient in order to ensure the existence of mild solutions tothe SPDE (44) Now we will prove that the existence of mildsolutions follows from global Lipschitz and linear growthconditions on 120572 and 120590 For this we recall an auxiliary resultwhich extends the Ito isometry (39)
Lemma 49 Let 119879 ge 0 be arbitrary and let 119883 = (119883119905)119905isin[0119879]
for all 119905 ge 0 and all ℎ isin 119867 Then for every F0-measurable
random variable ℎ0
Ω rarr 119867 there exists a (up toindistinguishability) unique mild solution 119883 to (44)
Proof The uniqueness of mild solutions to (44) is a directconsequence of Theorem 48 and hence we may concentrateon the existence proof which we divide into the followingseveral steps
Step 1 First we suppose that the initial condition ℎ0satisfies
E[ℎ02119901
] lt infin for some 119901 gt 1 Let 119879 ge 0 be arbitrary Wedefine the Banach space
1198712119901
119879(119867) = 119871
2119901
(Ω times [0 119879] P119879P otimes 119889119905119867) (129)
and prove that the variation of constants equation
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(130)
has a unique solution in the space 1198712119901
119879(119867) This is done in the
following three steps
Step 11 For 119883 isin 1198712119901
119879(119867) we define the process Φ119883 by
(Φ119883)119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(131)
Then the process Φ119883 is well defined Indeed by the growthestimate (9) the linear growth condition (127) and Holderrsquosinequality we have
Consequently there exists an index 119899 isin N such that Φ119899 is
a contraction and hence according to the extension of theBanach fixed point theorem (see Corollary 46) the mappingΦ has a unique fixed point 119883 isin 119871
2119901
119879(119867) This fixed point 119883 is
a solution to the variation of constants equation (130) Since119879 ge 0 was arbitrary there exists a process 119883 which is a solu-tion of the variation of constants equation
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
119905 ge 0
(146)
Step 14 In order to prove that 119883 is a mild solution to (44)it remains to show that 119883 has a continuous version ByLemma 12 the process
119905 997891997888rarr 119878119905ℎ0 119905 ge 0 (147)
is continuous and by Proposition 41 the process
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 119905 ge 0 (148)
is continuous too Moreover for every 119879 ge 0 we have by thelinear growth condition (128) Holderrsquos inequality and since119883 isin 119871
Thus by Proposition 43 the stochastic convolution 120590 ⋆ 119882
given by
(120590 ⋆ 119882)119905= int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0 (150)
has a continuous version and consequently the process119883hasa continuous version too This continuous version is a mildsolution to (44)
Step 2 Now let ℎ0 Ω rarr 119867 be an arbitraryF
0-measurable
random variable We define the sequence (ℎ119899)119899isinN of F
0-
measurable random variables as
ℎ119899
0= ℎ
01ℎ0le119899
119899 isin N (151)
Let 119899 isin N be arbitrary Then as ℎ119899
0is bounded we have
E[ℎ119899
02119901
] lt infin for all 119901 gt 1 By Step 1 the SPDE
119889119883119899
119905= (119860119883
119899
119905+ 120572 (119905 119883
119899
119905)) 119889119905 + 120590 (119905 119883
119899
119905) 119889119882
119905
119883119899
0= ℎ
119899
0
(152)
18 International Journal of Stochastic Analysis
has amild solution119883119899We define the sequence (Ω
119899)119899isinN sub F
0
as
Ω119899=
1003817100381710038171003817ℎ01003817100381710038171003817 le 119899 119899 isin N (153)
Then we have Ω119899sub Ω
119898for 119899 le 119898 Ω = ⋃
119899isinN Ω119899 and
Ω119899sub ℎ
119899
0= ℎ
119898
0 sub ℎ
119899
0= ℎ
0 forall119899 le 119898 (154)
Thus byTheorem 48 we have (up to indistinguishability)
119883119899
1Ω119899
= 119883119898
1Ω119899
forall119899 le 119898 (155)
Consequently the process
119883 = lim119899rarrinfin
119883119899
1Ω119899
(156)
is a well-defined continuous and adapted process and wehave
119883119899
1Ω119899
= 119883119898
1Ω119899
= 1198831Ω119899
forall119899 le 119898 (157)
Furthermore we obtain P-almost surely
119883119905= lim
119899rarrinfin
119883119899
1199051Ω119899
= lim119899rarrinfin
1Ω119899
(119878119905ℎ119899
0+ int
119905
0
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+int
119905
0
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ119899
0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= lim119899rarrinfin
1Ω119899
(119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= 119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(158)
proving that 119883 is a mild solution to (44)
Remark 51 For the proof of Theorem 50 we have usedCorollary 46 which is a slight extension of the Banach fixedpoint theorem Such an idea has been applied for example in[15]
Remark 52 A recent method for proving existence anduniqueness of mild solutions to the SPDE (44) is the methodof the moving frame presented in [6] see also [8] It allowsto reduce SPDE problems to the study of SDEs in infinitedimension In order to apply this method we need that thesemigroup (119878
119905)119905ge0
is a semigroup of pseudocontractions
We close this section with a consequence about theexistence of weak solutions
Corollary 53 Suppose that conditions (125)ndash(128) are ful-filled Let ℎ
0 Ω rarr 119867 be aF
0-measurable random variable
such that E[ℎ02119901
] lt infin for some 119901 gt 1 Then there exists a(up to indistinguishability) unique weak solution 119883 to (44)
Proof According to Proposition 35 every weak solution119883 to(44) is also a mild solution to (44)Therefore the uniquenessof weak solutions to (44) is a consequence of Theorem 48
It remains to prove the existence of a weak solution to(44) Let 119879 ge 0 be arbitrary By Theorem 50 and its proofthere exists a mild solution119883 isin 119871
2119901
119879(119867) to (44) By the linear
growth condition (128) and Holderrsquos inequality we obtain
showing that condition (72) is fulfilled Thus byProposition 37 the process 119883 is also a weak solution to(44)
8 Invariant Manifolds for WeakSolutions to SPDEs
In this section we deal with invariant manifolds for time-homogeneous SPDEs of the type (44) This topic arises fromthe natural desire to express the solutions of the SPDE (44)which generally live in the infinite dimensional Hilbert space119867 by means of a finite dimensional state process and thusto ensure larger analytical tractability Our goal is to findconditions on the generator 119860 and the coefficients 120572 120590 suchthat for every starting point of a finite dimensional submani-fold the solution process stays on the submanifold
We start with the required preliminaries about finitedimensional submanifolds inHilbert spaces In the sequel let119867 be a separable Hilbert space
Definition 54 Let 119898 119896 isin N be positive integers A subsetM sub 119867 is called an 119898-dimensional 119862119896-submanifold of 119867
International Journal of Stochastic Analysis 19
if for every ℎ isin M there exist an open neighborhood 119880 sub 119867
of ℎ an open set 119881 sub R119898 and a mapping 120601 isin 1198622
(119881119867) suchthat
(1) the mapping 120601 119881 rarr 119880 cap M is a homeomorphism(2) for all 119910 isin 119881 the mapping 119863120601(119910) is injective
The mapping 120601 is called a parametrization ofM around ℎ
In what follows let M be an 119898-dimensional 119862119896-
submanifold of 119867
Lemma 55 Let 120601119894
119881119894
rarr 119880119894cap M 119894 = 1 2 be two
parametrizations with119882= 1198801cap119880
2capM = 0Then themapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (160)
is a 119862119896-diffeomorphism
Proof See [16 Lemma 611]
Corollary 56 Let ℎ isin M be arbitrary and let 120601119894
119881119894
rarr
119880119894capM 119894 = 1 2 be two parametrizations ofM around ℎ Then
one has
1198631206011(119910
1) (R
119898
) = 1198631206012(119910
2) (R
119898
) (161)
where 119910119894= 120601
minus1
119894(ℎ) for 119894 = 1 2
Proof Since1198801and119880
2are open neighborhoods of ℎ we have
119882= 1198801cap 119880
2cap M = 0 Thus by Lemma 55 the mapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (162)
is a 119862119896-diffeomorphism Using the chain rule we obtain
1198631206012(119910
2) (R
119898
)
= 119863 (1206011∘ (120601
minus1
1∘ 120601
2)) (119910
2) (R
119898
)
= 1198631206011(119910
1)119863 (120601
minus1
1∘ 120601
2) (119910
2) (R
119898
)
sub 1198631206011(119910
1) (R
119898
)
(163)
and analogously we prove that 1198631206011(119910
1)(R119898
) sub
1198631206012(119910
2)(R119898
)
Definition 57 Let ℎ isin M be arbitraryThe tangent space ofMto ℎ is the subspace
119879ℎM = 119863120601 (119910) (R
119898
) (164)
where 119910 = 120601minus1
(ℎ) and 120601 119881 rarr 119880 cap M denotes aparametrization ofM around ℎ
Remark 58 Note that according to Corollary 56 theDefinition 57 of the tangent space 119879
ℎM does not depend on
the choice of the parametrization 120601 119881 rarr 119880 cap M
Proposition 59 Let ℎ isin M be arbitrary and let 120601 119881 rarr
119880 cap M be a parametrization ofM around ℎ Then there existan open set 119881
0sub 119881 an open neighborhood 119880
0sub 119880 of ℎ and
a mapping 120601 isin 119862119896
119887(R119898
119867) with 120601|1198810
= 120601|1198810
such that 120601|1198810
1198810
rarr 1198800cap M is a parametrization ofM around ℎ too
Proof See [16 Remark 611]
Remark 60 By Proposition 59 we may assume that anyparametrization 120601 119881 rarr 119880 cap M has an extension 120601 isin
119862119896
119887(R119898
119867)
Proposition 61 Let 119863 sub 119867 be a dense subset For every ℎ0isin
M there exist 1205771 120577
119898isin 119863 and a parametrization 120601 119881 rarr
119880 cap M around ℎ0such that
120601 (⟨120577 ℎ⟩) = ℎ forallℎ isin 119880 cap M (165)
where one uses the notation ⟨120577 ℎ⟩ = (⟨1205771 ℎ⟩ ⟨120577
119898 ℎ⟩) isin
R119898
Proof See [16 Proposition 612]
Proposition 62 Let 120601 119881 rarr 119880 cap M be a parametrizationas in Proposition 61 Then the following statements are true
(1) The elements 1205771 120577
119898are linearly independent in 119867
(2) For every ℎ isin 119880 cap M one has the direct sum decom-position
Therefore by Proposition 64 for every 1199100isin R119898 there exists
a (up to indistinguishability) unique weak solution to (192)which according to Proposition 39 is also a strong solutionto (192) The uniqueness of strong solutions to (192) is a con-sequence of Proposition 39 andTheorem 48
Now we are ready to formulate and prove ourmain resultof this section
Theorem 68 The following statements are equivalent
(1) The submanifoldM is locally invariant for (176)(2) One has
M sub D (119860) (195)
120590119895
(ℎ) isin 119879ℎM forallℎ isin M 119886119897119897 119895 isin N (196)
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) isin 119879ℎM forallℎ isin M (197)
(3) The operator 119860 is continuous onM and for each ℎ0isin
M there exists a local strong solution 119883 to (176) withsome lifetime 120591 gt 0 such that
119883119905and120591
isin M forall119905 ge 0 P-119886119897119898119900119904119905 119904119906119903119890119897119910 (198)
Proof (1)rArr(2) Let ℎ isin M be arbitrary By Proposition 61and Remark 60 there exist elements 120577
1 120577
119898isin D(119860
lowast
) anda parametrization 120601 119881 rarr 119880 cap M around ℎ such that theinverse 120601
minus1
119880 cap M rarr 119881 is given by 120601minus1
= ⟨120577 ∙⟩ and 120601
has an extension 120601 isin 1198622
119887(R119898
119867) Since the submanifoldM islocally invariant for (176) there exists a local weak solution119883
to (176) with initial condition ℎ and some lifetime 984858 gt 0 suchthat
119883119905and984858
isin M forall119905 ge 0 P-almost surely (199)
Since119880 is an open neighborhood of ℎ there exists 120598 gt 0 suchthat 119861
120598(ℎ) sub 119880 where 119861
120598(ℎ) denotes the open ball
119861120598(ℎ) = 119892 isin 119867
1003817100381710038171003817119892 minus ℎ1003817100381710038171003817 lt 120598 (200)
We define the stopping time
120591 = 984858 and inf 119905 ge 0 119883119905notin 119861
120598(ℎ) (201)
Since the process119883 has continuous sample paths and satisfies1198830= ℎ we have 120591 gt 0 and P-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (202)
22 International Journal of Stochastic Analysis
Defining the R119898-valued process 119884 = ⟨120577 119883⟩ we have P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (203)
Moreover since 119883 is a weak solution to (176) with initialcondition ℎ setting 119910 = ⟨120577 ℎ⟩ isin 119881 we have P-almost surely
119884119905and120591
= ⟨120577 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120577 120590119895
(119883119904)⟩ 119889120573
119895
119904
= ⟨120577 ℎ⟩ + int
119905and120591
0
120572120601120577
(⟨120577 119883119904⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(⟨120577 119883
119904⟩) 119889120573
119895
119904
= 119910 + int
119905and120591
0
120572120601120577
(119884119904) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(204)
showing that 119884 is a local strong solution to (192) with initialcondition 119910 By Itorsquos formula (Theorem 26) we obtain P-almost surely
119883119905and120591
= 120601 (119884119905and120591
)
= ℎ + int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904)))119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(205)
Now let 120585 isin D(119860lowast
) be arbitrary Then we have P-almostsurely
⟨120585 119883119905and120591
⟩
= ⟨120585 ℎ⟩
+ int
119905and120591
0
⟨120585119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 119863120601 (119884119904) 120590
119895
120601120577(119884
119904)⟩119889120573
119895
119904 119905 ge 0
(206)
On the other hand since 119883 is a local weak solution to (176)with initial condition ℎ and lifetime 120591 we have P-almostsurely for all 119905 ge 0 the identity
⟨120585 119883119905and120591
⟩ = ⟨120585 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904)⟩ 119889120573
119895
119904
(207)
Combining (206) and (207) yields up to indistinguishability
119861 + 119872 = 0 (208)
where the processes 119861 and 119872 are defined as
119861119905= int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩) 119889119904
119905 ge 0
119872119905= sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩ 119889120573
119895
119904
119905 ge 0
(209)
The process 119861 + 119872 is a continuous semimartingale withcanonical decomposition (208) Since the canonical decom-position of a continuous semimartingale is unique up toindistinguishability we deduce that 119861 = 119872 = 0 up toindistinguishability Using the Ito isometry (39) we obtain P-almost surely
int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩
+ ⟨120585 120572 (119883119904) minus 119863120601 (119884
showing that119883 is a local strong solution to (44) with lifetime120591
(3)rArr(1) This implication is a direct consequence ofProposition 29
The results from this section are closely related to the exis-tence of finite dimensional realizations that is the existenceof invariantmanifolds for each starting point ℎ
0 andwe point
out the papers [17ndash22] regarding this topic Furthermore wemention thatTheorem 68 has been extended in [23] to SPDEswith jumps
Acknowledgments
The author is grateful to Daniel Gaigall Georg GrafendorferFlorian Modler and Thomas Salfeld for the valuable com-ments and discussions
References
[1] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions CambridgeUniversity Press CambridgeUK 1992
[2] C Prevot and M Rockner A Concise Course on StochasticPartial Differential Equations Springer Berlin Germany 2007
[3] L Gawarecki and V Mandrekar Stochastic Differential Equa-tions in Infinite Dimensions with Applications to StochasticPartial Differential Equations Springer Berlin Germany 2011
[4] S Peszat and J Zabczyk Stochastic Partial Differential Equationswith Levy Noise Cambridge University Press Cambridge UK2007
[5] S Albeverio V Mandrekar and B Rudiger ldquoExistence of mildsolutions for stochastic differential equations and semilinearequations with non-Gaussian Levy noiserdquo Stochastic Processesand Their Applications vol 119 no 3 pp 835ndash863 2009
[6] D Filipovic S Tappe and J Teichmann ldquoJump-diffusions inHilbert spaces existence stability and numericsrdquo Stochasticsvol 82 no 5 pp 475ndash520 2010
[7] C Marinelli C Prevot and M Rockner ldquoRegular dependenceon initial data for stochastic evolution equations with multi-plicative Poisson noiserdquo Journal of Functional Analysis vol 258no 2 pp 616ndash649 2010
[8] S Tappe ldquoSome refinements of existence results for SPDEsdriven by Wiener processes and Poisson random measuresrdquoInternational Journal of Stochastic Analysis vol 2012 Article ID236327 24 pages 2012
[9] T Nakayama ldquoSupport theorem for mild solutions of SDErsquos inHilbert spacesrdquo Journal of Mathematical Sciences the Universityof Tokyo vol 11 no 3 pp 245ndash311 2004
[10] T Nakayama ldquoViability theorem for SPDErsquos including HJMframeworkrdquo Journal of Mathematical Sciences the University ofTokyo vol 11 no 3 pp 313ndash324 2004
[11] W Rudin Functional Analysis McGraw-Hill New York NYUSA 2nd edition 1991
[12] DWerner Funktionalanalysis Springer Berlin Germany 2007[13] C D Aliprantis and K C Border Infinite Dimensional Analysis
Springer Berlin Germany 3rd edition 2006[14] S N Ethier and T G KurtzMarkov Processes Characterization
andConvergenceWiley Series in Probability and Statistics JohnWiley amp Sons New Jersey NJ USA 1986
[15] V Mandrekar and B Rudiger ldquoExistence and uniqueness ofpath wise solutions for stochastic integral equations driven byLevy noise on separable Banach spacesrdquo Stochastics vol 78 no4 pp 189ndash212 2006
[16] D Filipovic Consistency Problems for Heath-Jarrow-MortonInterest Rate Models Springer Berlin Germany 2001
[17] T Bjork and L Svensson ldquoOn the existence of finite-dimensional realizations for nonlinear forward rate modelsrdquoMathematical Finance vol 11 no 2 pp 205ndash243 2001
[18] T Bjork and C Landen ldquoOn the construction of finite dimen-sional realizations for nonlinear forward rate modelsrdquo Financeand Stochastics vol 6 no 3 pp 303ndash331 2002
[19] D Filipovic and J Teichmann ldquoExistence of invariantmanifoldsfor stochastic equations in infinite dimensionrdquo Journal ofFunctional Analysis vol 197 no 2 pp 398ndash432 2003
[20] D Filipovic and J Teichmann ldquoOn the geometry of the termstructure of interest ratesrdquo Proceedings of The Royal Society ofLondon A vol 460 no 2041 pp 129ndash167 2004
[21] S Tappe ldquoAn alternative approach on the existence of affinerealizations for HJM term structure modelsrdquo Proceedings of
International Journal of Stochastic Analysis 25
TheRoyal Society of LondonA vol 466 no 2122 pp 3033ndash30602010
[22] S Tappe ldquoExistence of affine realizations for Levy term struc-ture modelsrdquo Proceedings of The Royal Society of London A vol468 no 2147 pp 3685ndash3704 2012
[23] D Filipovic S Tappe and J Teichmann ldquoInvariant manifoldswith boundary for jump-diffusionsrdquo httparxivorgabs12021076
(2) An element 119909 isin 119864 is called a fixed point of Φ if onehas
Φ (119909) = 119909 (109)
The following result is the well-known Banach fixed pointtheorem Its proof can be found for example in [13Theorem348]
Theorem 45 (The Banach fixed point theorem) Let 119864 be acomplete metric space and let Φ 119864 rarr 119864 be a contractionThen the mapping Φ has a unique fixed point
In this text we will use the following slight extension ofthe Banach fixed point theorem
Corollary 46 Let 119864 be a complete metric space and let Φ
119864 rarr 119864 be a mapping such that for some 119899 isin N the mappingΦ119899 is a contraction Then the mapping Φ has a unique fixed
point
Proof According to the Banach fixed point theorem(Theorem 45) the mapping Φ
119899 has a unique fixed point thatis there exists a unique element 119909 isin 119864 such that Φ119899
(119909) = 119909Therefore we have
Φ (119909) = Φ (Φ119899
(119909)) = Φ119899
(Φ (119909)) (110)
showing thatΦ(119909) is a fixedpoint ofΦ119899 SinceΦ119899 has a uniquefixed point we deduce thatΦ(119909) = 119909 showing that119909 is a fixedpoint of Φ
In order to prove uniqueness let 119910 isin 119864 be another fixedpoint ofΦ that is we haveΦ(119910) = 119910 By induction we obtain
showing that 119910 is a fixed point of Φ119899 Since the mapping Φ119899
has exactly one fixed point we obtain 119909 = 119910
An indispensable tool for proving uniqueness of mildsolutions to (44) will be the following version of Gronwallrsquosinequality see for example [14 Theorem 51]
Lemma 47 (Gronwallrsquos inequality) Let 119879 ge 0 be fixed let119891 [0 119879] rarr R
and hence by the Cauchy-Schwarz inequality the Ito isom-etry (39) the growth estimate (9) from Lemma 10 and thelocal Lipschitz conditions (113) we obtain
Using Gronwallrsquos inequality (see Lemma 47) we deduce that119891 equiv 0 Thus by the continuity of the sample paths of 119883 and119884 we obtain
P(⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ) = 1 forall119899 isin N (120)
and hence by the continuity of the probability measureP weconclude that
P(⋂
119905ge0
119883119905and120591
1Γ= 119884
119905and1205911Γ)
= P(⋂
119899isinN
⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ)
= lim119899rarrinfin
P(⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ) = 1
(121)
which completes the proof
The local Lipschitz conditions (113) are in general notsufficient in order to ensure the existence of mild solutions tothe SPDE (44) Now we will prove that the existence of mildsolutions follows from global Lipschitz and linear growthconditions on 120572 and 120590 For this we recall an auxiliary resultwhich extends the Ito isometry (39)
Lemma 49 Let 119879 ge 0 be arbitrary and let 119883 = (119883119905)119905isin[0119879]
for all 119905 ge 0 and all ℎ isin 119867 Then for every F0-measurable
random variable ℎ0
Ω rarr 119867 there exists a (up toindistinguishability) unique mild solution 119883 to (44)
Proof The uniqueness of mild solutions to (44) is a directconsequence of Theorem 48 and hence we may concentrateon the existence proof which we divide into the followingseveral steps
Step 1 First we suppose that the initial condition ℎ0satisfies
E[ℎ02119901
] lt infin for some 119901 gt 1 Let 119879 ge 0 be arbitrary Wedefine the Banach space
1198712119901
119879(119867) = 119871
2119901
(Ω times [0 119879] P119879P otimes 119889119905119867) (129)
and prove that the variation of constants equation
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(130)
has a unique solution in the space 1198712119901
119879(119867) This is done in the
following three steps
Step 11 For 119883 isin 1198712119901
119879(119867) we define the process Φ119883 by
(Φ119883)119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(131)
Then the process Φ119883 is well defined Indeed by the growthestimate (9) the linear growth condition (127) and Holderrsquosinequality we have
Consequently there exists an index 119899 isin N such that Φ119899 is
a contraction and hence according to the extension of theBanach fixed point theorem (see Corollary 46) the mappingΦ has a unique fixed point 119883 isin 119871
2119901
119879(119867) This fixed point 119883 is
a solution to the variation of constants equation (130) Since119879 ge 0 was arbitrary there exists a process 119883 which is a solu-tion of the variation of constants equation
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
119905 ge 0
(146)
Step 14 In order to prove that 119883 is a mild solution to (44)it remains to show that 119883 has a continuous version ByLemma 12 the process
119905 997891997888rarr 119878119905ℎ0 119905 ge 0 (147)
is continuous and by Proposition 41 the process
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 119905 ge 0 (148)
is continuous too Moreover for every 119879 ge 0 we have by thelinear growth condition (128) Holderrsquos inequality and since119883 isin 119871
Thus by Proposition 43 the stochastic convolution 120590 ⋆ 119882
given by
(120590 ⋆ 119882)119905= int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0 (150)
has a continuous version and consequently the process119883hasa continuous version too This continuous version is a mildsolution to (44)
Step 2 Now let ℎ0 Ω rarr 119867 be an arbitraryF
0-measurable
random variable We define the sequence (ℎ119899)119899isinN of F
0-
measurable random variables as
ℎ119899
0= ℎ
01ℎ0le119899
119899 isin N (151)
Let 119899 isin N be arbitrary Then as ℎ119899
0is bounded we have
E[ℎ119899
02119901
] lt infin for all 119901 gt 1 By Step 1 the SPDE
119889119883119899
119905= (119860119883
119899
119905+ 120572 (119905 119883
119899
119905)) 119889119905 + 120590 (119905 119883
119899
119905) 119889119882
119905
119883119899
0= ℎ
119899
0
(152)
18 International Journal of Stochastic Analysis
has amild solution119883119899We define the sequence (Ω
119899)119899isinN sub F
0
as
Ω119899=
1003817100381710038171003817ℎ01003817100381710038171003817 le 119899 119899 isin N (153)
Then we have Ω119899sub Ω
119898for 119899 le 119898 Ω = ⋃
119899isinN Ω119899 and
Ω119899sub ℎ
119899
0= ℎ
119898
0 sub ℎ
119899
0= ℎ
0 forall119899 le 119898 (154)
Thus byTheorem 48 we have (up to indistinguishability)
119883119899
1Ω119899
= 119883119898
1Ω119899
forall119899 le 119898 (155)
Consequently the process
119883 = lim119899rarrinfin
119883119899
1Ω119899
(156)
is a well-defined continuous and adapted process and wehave
119883119899
1Ω119899
= 119883119898
1Ω119899
= 1198831Ω119899
forall119899 le 119898 (157)
Furthermore we obtain P-almost surely
119883119905= lim
119899rarrinfin
119883119899
1199051Ω119899
= lim119899rarrinfin
1Ω119899
(119878119905ℎ119899
0+ int
119905
0
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+int
119905
0
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ119899
0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= lim119899rarrinfin
1Ω119899
(119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= 119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(158)
proving that 119883 is a mild solution to (44)
Remark 51 For the proof of Theorem 50 we have usedCorollary 46 which is a slight extension of the Banach fixedpoint theorem Such an idea has been applied for example in[15]
Remark 52 A recent method for proving existence anduniqueness of mild solutions to the SPDE (44) is the methodof the moving frame presented in [6] see also [8] It allowsto reduce SPDE problems to the study of SDEs in infinitedimension In order to apply this method we need that thesemigroup (119878
119905)119905ge0
is a semigroup of pseudocontractions
We close this section with a consequence about theexistence of weak solutions
Corollary 53 Suppose that conditions (125)ndash(128) are ful-filled Let ℎ
0 Ω rarr 119867 be aF
0-measurable random variable
such that E[ℎ02119901
] lt infin for some 119901 gt 1 Then there exists a(up to indistinguishability) unique weak solution 119883 to (44)
Proof According to Proposition 35 every weak solution119883 to(44) is also a mild solution to (44)Therefore the uniquenessof weak solutions to (44) is a consequence of Theorem 48
It remains to prove the existence of a weak solution to(44) Let 119879 ge 0 be arbitrary By Theorem 50 and its proofthere exists a mild solution119883 isin 119871
2119901
119879(119867) to (44) By the linear
growth condition (128) and Holderrsquos inequality we obtain
showing that condition (72) is fulfilled Thus byProposition 37 the process 119883 is also a weak solution to(44)
8 Invariant Manifolds for WeakSolutions to SPDEs
In this section we deal with invariant manifolds for time-homogeneous SPDEs of the type (44) This topic arises fromthe natural desire to express the solutions of the SPDE (44)which generally live in the infinite dimensional Hilbert space119867 by means of a finite dimensional state process and thusto ensure larger analytical tractability Our goal is to findconditions on the generator 119860 and the coefficients 120572 120590 suchthat for every starting point of a finite dimensional submani-fold the solution process stays on the submanifold
We start with the required preliminaries about finitedimensional submanifolds inHilbert spaces In the sequel let119867 be a separable Hilbert space
Definition 54 Let 119898 119896 isin N be positive integers A subsetM sub 119867 is called an 119898-dimensional 119862119896-submanifold of 119867
International Journal of Stochastic Analysis 19
if for every ℎ isin M there exist an open neighborhood 119880 sub 119867
of ℎ an open set 119881 sub R119898 and a mapping 120601 isin 1198622
(119881119867) suchthat
(1) the mapping 120601 119881 rarr 119880 cap M is a homeomorphism(2) for all 119910 isin 119881 the mapping 119863120601(119910) is injective
The mapping 120601 is called a parametrization ofM around ℎ
In what follows let M be an 119898-dimensional 119862119896-
submanifold of 119867
Lemma 55 Let 120601119894
119881119894
rarr 119880119894cap M 119894 = 1 2 be two
parametrizations with119882= 1198801cap119880
2capM = 0Then themapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (160)
is a 119862119896-diffeomorphism
Proof See [16 Lemma 611]
Corollary 56 Let ℎ isin M be arbitrary and let 120601119894
119881119894
rarr
119880119894capM 119894 = 1 2 be two parametrizations ofM around ℎ Then
one has
1198631206011(119910
1) (R
119898
) = 1198631206012(119910
2) (R
119898
) (161)
where 119910119894= 120601
minus1
119894(ℎ) for 119894 = 1 2
Proof Since1198801and119880
2are open neighborhoods of ℎ we have
119882= 1198801cap 119880
2cap M = 0 Thus by Lemma 55 the mapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (162)
is a 119862119896-diffeomorphism Using the chain rule we obtain
1198631206012(119910
2) (R
119898
)
= 119863 (1206011∘ (120601
minus1
1∘ 120601
2)) (119910
2) (R
119898
)
= 1198631206011(119910
1)119863 (120601
minus1
1∘ 120601
2) (119910
2) (R
119898
)
sub 1198631206011(119910
1) (R
119898
)
(163)
and analogously we prove that 1198631206011(119910
1)(R119898
) sub
1198631206012(119910
2)(R119898
)
Definition 57 Let ℎ isin M be arbitraryThe tangent space ofMto ℎ is the subspace
119879ℎM = 119863120601 (119910) (R
119898
) (164)
where 119910 = 120601minus1
(ℎ) and 120601 119881 rarr 119880 cap M denotes aparametrization ofM around ℎ
Remark 58 Note that according to Corollary 56 theDefinition 57 of the tangent space 119879
ℎM does not depend on
the choice of the parametrization 120601 119881 rarr 119880 cap M
Proposition 59 Let ℎ isin M be arbitrary and let 120601 119881 rarr
119880 cap M be a parametrization ofM around ℎ Then there existan open set 119881
0sub 119881 an open neighborhood 119880
0sub 119880 of ℎ and
a mapping 120601 isin 119862119896
119887(R119898
119867) with 120601|1198810
= 120601|1198810
such that 120601|1198810
1198810
rarr 1198800cap M is a parametrization ofM around ℎ too
Proof See [16 Remark 611]
Remark 60 By Proposition 59 we may assume that anyparametrization 120601 119881 rarr 119880 cap M has an extension 120601 isin
119862119896
119887(R119898
119867)
Proposition 61 Let 119863 sub 119867 be a dense subset For every ℎ0isin
M there exist 1205771 120577
119898isin 119863 and a parametrization 120601 119881 rarr
119880 cap M around ℎ0such that
120601 (⟨120577 ℎ⟩) = ℎ forallℎ isin 119880 cap M (165)
where one uses the notation ⟨120577 ℎ⟩ = (⟨1205771 ℎ⟩ ⟨120577
119898 ℎ⟩) isin
R119898
Proof See [16 Proposition 612]
Proposition 62 Let 120601 119881 rarr 119880 cap M be a parametrizationas in Proposition 61 Then the following statements are true
(1) The elements 1205771 120577
119898are linearly independent in 119867
(2) For every ℎ isin 119880 cap M one has the direct sum decom-position
Therefore by Proposition 64 for every 1199100isin R119898 there exists
a (up to indistinguishability) unique weak solution to (192)which according to Proposition 39 is also a strong solutionto (192) The uniqueness of strong solutions to (192) is a con-sequence of Proposition 39 andTheorem 48
Now we are ready to formulate and prove ourmain resultof this section
Theorem 68 The following statements are equivalent
(1) The submanifoldM is locally invariant for (176)(2) One has
M sub D (119860) (195)
120590119895
(ℎ) isin 119879ℎM forallℎ isin M 119886119897119897 119895 isin N (196)
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) isin 119879ℎM forallℎ isin M (197)
(3) The operator 119860 is continuous onM and for each ℎ0isin
M there exists a local strong solution 119883 to (176) withsome lifetime 120591 gt 0 such that
119883119905and120591
isin M forall119905 ge 0 P-119886119897119898119900119904119905 119904119906119903119890119897119910 (198)
Proof (1)rArr(2) Let ℎ isin M be arbitrary By Proposition 61and Remark 60 there exist elements 120577
1 120577
119898isin D(119860
lowast
) anda parametrization 120601 119881 rarr 119880 cap M around ℎ such that theinverse 120601
minus1
119880 cap M rarr 119881 is given by 120601minus1
= ⟨120577 ∙⟩ and 120601
has an extension 120601 isin 1198622
119887(R119898
119867) Since the submanifoldM islocally invariant for (176) there exists a local weak solution119883
to (176) with initial condition ℎ and some lifetime 984858 gt 0 suchthat
119883119905and984858
isin M forall119905 ge 0 P-almost surely (199)
Since119880 is an open neighborhood of ℎ there exists 120598 gt 0 suchthat 119861
120598(ℎ) sub 119880 where 119861
120598(ℎ) denotes the open ball
119861120598(ℎ) = 119892 isin 119867
1003817100381710038171003817119892 minus ℎ1003817100381710038171003817 lt 120598 (200)
We define the stopping time
120591 = 984858 and inf 119905 ge 0 119883119905notin 119861
120598(ℎ) (201)
Since the process119883 has continuous sample paths and satisfies1198830= ℎ we have 120591 gt 0 and P-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (202)
22 International Journal of Stochastic Analysis
Defining the R119898-valued process 119884 = ⟨120577 119883⟩ we have P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (203)
Moreover since 119883 is a weak solution to (176) with initialcondition ℎ setting 119910 = ⟨120577 ℎ⟩ isin 119881 we have P-almost surely
119884119905and120591
= ⟨120577 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120577 120590119895
(119883119904)⟩ 119889120573
119895
119904
= ⟨120577 ℎ⟩ + int
119905and120591
0
120572120601120577
(⟨120577 119883119904⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(⟨120577 119883
119904⟩) 119889120573
119895
119904
= 119910 + int
119905and120591
0
120572120601120577
(119884119904) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(204)
showing that 119884 is a local strong solution to (192) with initialcondition 119910 By Itorsquos formula (Theorem 26) we obtain P-almost surely
119883119905and120591
= 120601 (119884119905and120591
)
= ℎ + int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904)))119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(205)
Now let 120585 isin D(119860lowast
) be arbitrary Then we have P-almostsurely
⟨120585 119883119905and120591
⟩
= ⟨120585 ℎ⟩
+ int
119905and120591
0
⟨120585119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 119863120601 (119884119904) 120590
119895
120601120577(119884
119904)⟩119889120573
119895
119904 119905 ge 0
(206)
On the other hand since 119883 is a local weak solution to (176)with initial condition ℎ and lifetime 120591 we have P-almostsurely for all 119905 ge 0 the identity
⟨120585 119883119905and120591
⟩ = ⟨120585 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904)⟩ 119889120573
119895
119904
(207)
Combining (206) and (207) yields up to indistinguishability
119861 + 119872 = 0 (208)
where the processes 119861 and 119872 are defined as
119861119905= int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩) 119889119904
119905 ge 0
119872119905= sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩ 119889120573
119895
119904
119905 ge 0
(209)
The process 119861 + 119872 is a continuous semimartingale withcanonical decomposition (208) Since the canonical decom-position of a continuous semimartingale is unique up toindistinguishability we deduce that 119861 = 119872 = 0 up toindistinguishability Using the Ito isometry (39) we obtain P-almost surely
int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩
+ ⟨120585 120572 (119883119904) minus 119863120601 (119884
showing that119883 is a local strong solution to (44) with lifetime120591
(3)rArr(1) This implication is a direct consequence ofProposition 29
The results from this section are closely related to the exis-tence of finite dimensional realizations that is the existenceof invariantmanifolds for each starting point ℎ
0 andwe point
out the papers [17ndash22] regarding this topic Furthermore wemention thatTheorem 68 has been extended in [23] to SPDEswith jumps
Acknowledgments
The author is grateful to Daniel Gaigall Georg GrafendorferFlorian Modler and Thomas Salfeld for the valuable com-ments and discussions
References
[1] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions CambridgeUniversity Press CambridgeUK 1992
[2] C Prevot and M Rockner A Concise Course on StochasticPartial Differential Equations Springer Berlin Germany 2007
[3] L Gawarecki and V Mandrekar Stochastic Differential Equa-tions in Infinite Dimensions with Applications to StochasticPartial Differential Equations Springer Berlin Germany 2011
[4] S Peszat and J Zabczyk Stochastic Partial Differential Equationswith Levy Noise Cambridge University Press Cambridge UK2007
[5] S Albeverio V Mandrekar and B Rudiger ldquoExistence of mildsolutions for stochastic differential equations and semilinearequations with non-Gaussian Levy noiserdquo Stochastic Processesand Their Applications vol 119 no 3 pp 835ndash863 2009
[6] D Filipovic S Tappe and J Teichmann ldquoJump-diffusions inHilbert spaces existence stability and numericsrdquo Stochasticsvol 82 no 5 pp 475ndash520 2010
[7] C Marinelli C Prevot and M Rockner ldquoRegular dependenceon initial data for stochastic evolution equations with multi-plicative Poisson noiserdquo Journal of Functional Analysis vol 258no 2 pp 616ndash649 2010
[8] S Tappe ldquoSome refinements of existence results for SPDEsdriven by Wiener processes and Poisson random measuresrdquoInternational Journal of Stochastic Analysis vol 2012 Article ID236327 24 pages 2012
[9] T Nakayama ldquoSupport theorem for mild solutions of SDErsquos inHilbert spacesrdquo Journal of Mathematical Sciences the Universityof Tokyo vol 11 no 3 pp 245ndash311 2004
[10] T Nakayama ldquoViability theorem for SPDErsquos including HJMframeworkrdquo Journal of Mathematical Sciences the University ofTokyo vol 11 no 3 pp 313ndash324 2004
[11] W Rudin Functional Analysis McGraw-Hill New York NYUSA 2nd edition 1991
[12] DWerner Funktionalanalysis Springer Berlin Germany 2007[13] C D Aliprantis and K C Border Infinite Dimensional Analysis
Springer Berlin Germany 3rd edition 2006[14] S N Ethier and T G KurtzMarkov Processes Characterization
andConvergenceWiley Series in Probability and Statistics JohnWiley amp Sons New Jersey NJ USA 1986
[15] V Mandrekar and B Rudiger ldquoExistence and uniqueness ofpath wise solutions for stochastic integral equations driven byLevy noise on separable Banach spacesrdquo Stochastics vol 78 no4 pp 189ndash212 2006
[16] D Filipovic Consistency Problems for Heath-Jarrow-MortonInterest Rate Models Springer Berlin Germany 2001
[17] T Bjork and L Svensson ldquoOn the existence of finite-dimensional realizations for nonlinear forward rate modelsrdquoMathematical Finance vol 11 no 2 pp 205ndash243 2001
[18] T Bjork and C Landen ldquoOn the construction of finite dimen-sional realizations for nonlinear forward rate modelsrdquo Financeand Stochastics vol 6 no 3 pp 303ndash331 2002
[19] D Filipovic and J Teichmann ldquoExistence of invariantmanifoldsfor stochastic equations in infinite dimensionrdquo Journal ofFunctional Analysis vol 197 no 2 pp 398ndash432 2003
[20] D Filipovic and J Teichmann ldquoOn the geometry of the termstructure of interest ratesrdquo Proceedings of The Royal Society ofLondon A vol 460 no 2041 pp 129ndash167 2004
[21] S Tappe ldquoAn alternative approach on the existence of affinerealizations for HJM term structure modelsrdquo Proceedings of
International Journal of Stochastic Analysis 25
TheRoyal Society of LondonA vol 466 no 2122 pp 3033ndash30602010
[22] S Tappe ldquoExistence of affine realizations for Levy term struc-ture modelsrdquo Proceedings of The Royal Society of London A vol468 no 2147 pp 3685ndash3704 2012
[23] D Filipovic S Tappe and J Teichmann ldquoInvariant manifoldswith boundary for jump-diffusionsrdquo httparxivorgabs12021076
(2) An element 119909 isin 119864 is called a fixed point of Φ if onehas
Φ (119909) = 119909 (109)
The following result is the well-known Banach fixed pointtheorem Its proof can be found for example in [13Theorem348]
Theorem 45 (The Banach fixed point theorem) Let 119864 be acomplete metric space and let Φ 119864 rarr 119864 be a contractionThen the mapping Φ has a unique fixed point
In this text we will use the following slight extension ofthe Banach fixed point theorem
Corollary 46 Let 119864 be a complete metric space and let Φ
119864 rarr 119864 be a mapping such that for some 119899 isin N the mappingΦ119899 is a contraction Then the mapping Φ has a unique fixed
point
Proof According to the Banach fixed point theorem(Theorem 45) the mapping Φ
119899 has a unique fixed point thatis there exists a unique element 119909 isin 119864 such that Φ119899
(119909) = 119909Therefore we have
Φ (119909) = Φ (Φ119899
(119909)) = Φ119899
(Φ (119909)) (110)
showing thatΦ(119909) is a fixedpoint ofΦ119899 SinceΦ119899 has a uniquefixed point we deduce thatΦ(119909) = 119909 showing that119909 is a fixedpoint of Φ
In order to prove uniqueness let 119910 isin 119864 be another fixedpoint ofΦ that is we haveΦ(119910) = 119910 By induction we obtain
showing that 119910 is a fixed point of Φ119899 Since the mapping Φ119899
has exactly one fixed point we obtain 119909 = 119910
An indispensable tool for proving uniqueness of mildsolutions to (44) will be the following version of Gronwallrsquosinequality see for example [14 Theorem 51]
Lemma 47 (Gronwallrsquos inequality) Let 119879 ge 0 be fixed let119891 [0 119879] rarr R
and hence by the Cauchy-Schwarz inequality the Ito isom-etry (39) the growth estimate (9) from Lemma 10 and thelocal Lipschitz conditions (113) we obtain
Using Gronwallrsquos inequality (see Lemma 47) we deduce that119891 equiv 0 Thus by the continuity of the sample paths of 119883 and119884 we obtain
P(⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ) = 1 forall119899 isin N (120)
and hence by the continuity of the probability measureP weconclude that
P(⋂
119905ge0
119883119905and120591
1Γ= 119884
119905and1205911Γ)
= P(⋂
119899isinN
⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ)
= lim119899rarrinfin
P(⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ) = 1
(121)
which completes the proof
The local Lipschitz conditions (113) are in general notsufficient in order to ensure the existence of mild solutions tothe SPDE (44) Now we will prove that the existence of mildsolutions follows from global Lipschitz and linear growthconditions on 120572 and 120590 For this we recall an auxiliary resultwhich extends the Ito isometry (39)
Lemma 49 Let 119879 ge 0 be arbitrary and let 119883 = (119883119905)119905isin[0119879]
for all 119905 ge 0 and all ℎ isin 119867 Then for every F0-measurable
random variable ℎ0
Ω rarr 119867 there exists a (up toindistinguishability) unique mild solution 119883 to (44)
Proof The uniqueness of mild solutions to (44) is a directconsequence of Theorem 48 and hence we may concentrateon the existence proof which we divide into the followingseveral steps
Step 1 First we suppose that the initial condition ℎ0satisfies
E[ℎ02119901
] lt infin for some 119901 gt 1 Let 119879 ge 0 be arbitrary Wedefine the Banach space
1198712119901
119879(119867) = 119871
2119901
(Ω times [0 119879] P119879P otimes 119889119905119867) (129)
and prove that the variation of constants equation
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(130)
has a unique solution in the space 1198712119901
119879(119867) This is done in the
following three steps
Step 11 For 119883 isin 1198712119901
119879(119867) we define the process Φ119883 by
(Φ119883)119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(131)
Then the process Φ119883 is well defined Indeed by the growthestimate (9) the linear growth condition (127) and Holderrsquosinequality we have
Consequently there exists an index 119899 isin N such that Φ119899 is
a contraction and hence according to the extension of theBanach fixed point theorem (see Corollary 46) the mappingΦ has a unique fixed point 119883 isin 119871
2119901
119879(119867) This fixed point 119883 is
a solution to the variation of constants equation (130) Since119879 ge 0 was arbitrary there exists a process 119883 which is a solu-tion of the variation of constants equation
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
119905 ge 0
(146)
Step 14 In order to prove that 119883 is a mild solution to (44)it remains to show that 119883 has a continuous version ByLemma 12 the process
119905 997891997888rarr 119878119905ℎ0 119905 ge 0 (147)
is continuous and by Proposition 41 the process
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 119905 ge 0 (148)
is continuous too Moreover for every 119879 ge 0 we have by thelinear growth condition (128) Holderrsquos inequality and since119883 isin 119871
Thus by Proposition 43 the stochastic convolution 120590 ⋆ 119882
given by
(120590 ⋆ 119882)119905= int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0 (150)
has a continuous version and consequently the process119883hasa continuous version too This continuous version is a mildsolution to (44)
Step 2 Now let ℎ0 Ω rarr 119867 be an arbitraryF
0-measurable
random variable We define the sequence (ℎ119899)119899isinN of F
0-
measurable random variables as
ℎ119899
0= ℎ
01ℎ0le119899
119899 isin N (151)
Let 119899 isin N be arbitrary Then as ℎ119899
0is bounded we have
E[ℎ119899
02119901
] lt infin for all 119901 gt 1 By Step 1 the SPDE
119889119883119899
119905= (119860119883
119899
119905+ 120572 (119905 119883
119899
119905)) 119889119905 + 120590 (119905 119883
119899
119905) 119889119882
119905
119883119899
0= ℎ
119899
0
(152)
18 International Journal of Stochastic Analysis
has amild solution119883119899We define the sequence (Ω
119899)119899isinN sub F
0
as
Ω119899=
1003817100381710038171003817ℎ01003817100381710038171003817 le 119899 119899 isin N (153)
Then we have Ω119899sub Ω
119898for 119899 le 119898 Ω = ⋃
119899isinN Ω119899 and
Ω119899sub ℎ
119899
0= ℎ
119898
0 sub ℎ
119899
0= ℎ
0 forall119899 le 119898 (154)
Thus byTheorem 48 we have (up to indistinguishability)
119883119899
1Ω119899
= 119883119898
1Ω119899
forall119899 le 119898 (155)
Consequently the process
119883 = lim119899rarrinfin
119883119899
1Ω119899
(156)
is a well-defined continuous and adapted process and wehave
119883119899
1Ω119899
= 119883119898
1Ω119899
= 1198831Ω119899
forall119899 le 119898 (157)
Furthermore we obtain P-almost surely
119883119905= lim
119899rarrinfin
119883119899
1199051Ω119899
= lim119899rarrinfin
1Ω119899
(119878119905ℎ119899
0+ int
119905
0
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+int
119905
0
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ119899
0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= lim119899rarrinfin
1Ω119899
(119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= 119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(158)
proving that 119883 is a mild solution to (44)
Remark 51 For the proof of Theorem 50 we have usedCorollary 46 which is a slight extension of the Banach fixedpoint theorem Such an idea has been applied for example in[15]
Remark 52 A recent method for proving existence anduniqueness of mild solutions to the SPDE (44) is the methodof the moving frame presented in [6] see also [8] It allowsto reduce SPDE problems to the study of SDEs in infinitedimension In order to apply this method we need that thesemigroup (119878
119905)119905ge0
is a semigroup of pseudocontractions
We close this section with a consequence about theexistence of weak solutions
Corollary 53 Suppose that conditions (125)ndash(128) are ful-filled Let ℎ
0 Ω rarr 119867 be aF
0-measurable random variable
such that E[ℎ02119901
] lt infin for some 119901 gt 1 Then there exists a(up to indistinguishability) unique weak solution 119883 to (44)
Proof According to Proposition 35 every weak solution119883 to(44) is also a mild solution to (44)Therefore the uniquenessof weak solutions to (44) is a consequence of Theorem 48
It remains to prove the existence of a weak solution to(44) Let 119879 ge 0 be arbitrary By Theorem 50 and its proofthere exists a mild solution119883 isin 119871
2119901
119879(119867) to (44) By the linear
growth condition (128) and Holderrsquos inequality we obtain
showing that condition (72) is fulfilled Thus byProposition 37 the process 119883 is also a weak solution to(44)
8 Invariant Manifolds for WeakSolutions to SPDEs
In this section we deal with invariant manifolds for time-homogeneous SPDEs of the type (44) This topic arises fromthe natural desire to express the solutions of the SPDE (44)which generally live in the infinite dimensional Hilbert space119867 by means of a finite dimensional state process and thusto ensure larger analytical tractability Our goal is to findconditions on the generator 119860 and the coefficients 120572 120590 suchthat for every starting point of a finite dimensional submani-fold the solution process stays on the submanifold
We start with the required preliminaries about finitedimensional submanifolds inHilbert spaces In the sequel let119867 be a separable Hilbert space
Definition 54 Let 119898 119896 isin N be positive integers A subsetM sub 119867 is called an 119898-dimensional 119862119896-submanifold of 119867
International Journal of Stochastic Analysis 19
if for every ℎ isin M there exist an open neighborhood 119880 sub 119867
of ℎ an open set 119881 sub R119898 and a mapping 120601 isin 1198622
(119881119867) suchthat
(1) the mapping 120601 119881 rarr 119880 cap M is a homeomorphism(2) for all 119910 isin 119881 the mapping 119863120601(119910) is injective
The mapping 120601 is called a parametrization ofM around ℎ
In what follows let M be an 119898-dimensional 119862119896-
submanifold of 119867
Lemma 55 Let 120601119894
119881119894
rarr 119880119894cap M 119894 = 1 2 be two
parametrizations with119882= 1198801cap119880
2capM = 0Then themapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (160)
is a 119862119896-diffeomorphism
Proof See [16 Lemma 611]
Corollary 56 Let ℎ isin M be arbitrary and let 120601119894
119881119894
rarr
119880119894capM 119894 = 1 2 be two parametrizations ofM around ℎ Then
one has
1198631206011(119910
1) (R
119898
) = 1198631206012(119910
2) (R
119898
) (161)
where 119910119894= 120601
minus1
119894(ℎ) for 119894 = 1 2
Proof Since1198801and119880
2are open neighborhoods of ℎ we have
119882= 1198801cap 119880
2cap M = 0 Thus by Lemma 55 the mapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (162)
is a 119862119896-diffeomorphism Using the chain rule we obtain
1198631206012(119910
2) (R
119898
)
= 119863 (1206011∘ (120601
minus1
1∘ 120601
2)) (119910
2) (R
119898
)
= 1198631206011(119910
1)119863 (120601
minus1
1∘ 120601
2) (119910
2) (R
119898
)
sub 1198631206011(119910
1) (R
119898
)
(163)
and analogously we prove that 1198631206011(119910
1)(R119898
) sub
1198631206012(119910
2)(R119898
)
Definition 57 Let ℎ isin M be arbitraryThe tangent space ofMto ℎ is the subspace
119879ℎM = 119863120601 (119910) (R
119898
) (164)
where 119910 = 120601minus1
(ℎ) and 120601 119881 rarr 119880 cap M denotes aparametrization ofM around ℎ
Remark 58 Note that according to Corollary 56 theDefinition 57 of the tangent space 119879
ℎM does not depend on
the choice of the parametrization 120601 119881 rarr 119880 cap M
Proposition 59 Let ℎ isin M be arbitrary and let 120601 119881 rarr
119880 cap M be a parametrization ofM around ℎ Then there existan open set 119881
0sub 119881 an open neighborhood 119880
0sub 119880 of ℎ and
a mapping 120601 isin 119862119896
119887(R119898
119867) with 120601|1198810
= 120601|1198810
such that 120601|1198810
1198810
rarr 1198800cap M is a parametrization ofM around ℎ too
Proof See [16 Remark 611]
Remark 60 By Proposition 59 we may assume that anyparametrization 120601 119881 rarr 119880 cap M has an extension 120601 isin
119862119896
119887(R119898
119867)
Proposition 61 Let 119863 sub 119867 be a dense subset For every ℎ0isin
M there exist 1205771 120577
119898isin 119863 and a parametrization 120601 119881 rarr
119880 cap M around ℎ0such that
120601 (⟨120577 ℎ⟩) = ℎ forallℎ isin 119880 cap M (165)
where one uses the notation ⟨120577 ℎ⟩ = (⟨1205771 ℎ⟩ ⟨120577
119898 ℎ⟩) isin
R119898
Proof See [16 Proposition 612]
Proposition 62 Let 120601 119881 rarr 119880 cap M be a parametrizationas in Proposition 61 Then the following statements are true
(1) The elements 1205771 120577
119898are linearly independent in 119867
(2) For every ℎ isin 119880 cap M one has the direct sum decom-position
Therefore by Proposition 64 for every 1199100isin R119898 there exists
a (up to indistinguishability) unique weak solution to (192)which according to Proposition 39 is also a strong solutionto (192) The uniqueness of strong solutions to (192) is a con-sequence of Proposition 39 andTheorem 48
Now we are ready to formulate and prove ourmain resultof this section
Theorem 68 The following statements are equivalent
(1) The submanifoldM is locally invariant for (176)(2) One has
M sub D (119860) (195)
120590119895
(ℎ) isin 119879ℎM forallℎ isin M 119886119897119897 119895 isin N (196)
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) isin 119879ℎM forallℎ isin M (197)
(3) The operator 119860 is continuous onM and for each ℎ0isin
M there exists a local strong solution 119883 to (176) withsome lifetime 120591 gt 0 such that
119883119905and120591
isin M forall119905 ge 0 P-119886119897119898119900119904119905 119904119906119903119890119897119910 (198)
Proof (1)rArr(2) Let ℎ isin M be arbitrary By Proposition 61and Remark 60 there exist elements 120577
1 120577
119898isin D(119860
lowast
) anda parametrization 120601 119881 rarr 119880 cap M around ℎ such that theinverse 120601
minus1
119880 cap M rarr 119881 is given by 120601minus1
= ⟨120577 ∙⟩ and 120601
has an extension 120601 isin 1198622
119887(R119898
119867) Since the submanifoldM islocally invariant for (176) there exists a local weak solution119883
to (176) with initial condition ℎ and some lifetime 984858 gt 0 suchthat
119883119905and984858
isin M forall119905 ge 0 P-almost surely (199)
Since119880 is an open neighborhood of ℎ there exists 120598 gt 0 suchthat 119861
120598(ℎ) sub 119880 where 119861
120598(ℎ) denotes the open ball
119861120598(ℎ) = 119892 isin 119867
1003817100381710038171003817119892 minus ℎ1003817100381710038171003817 lt 120598 (200)
We define the stopping time
120591 = 984858 and inf 119905 ge 0 119883119905notin 119861
120598(ℎ) (201)
Since the process119883 has continuous sample paths and satisfies1198830= ℎ we have 120591 gt 0 and P-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (202)
22 International Journal of Stochastic Analysis
Defining the R119898-valued process 119884 = ⟨120577 119883⟩ we have P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (203)
Moreover since 119883 is a weak solution to (176) with initialcondition ℎ setting 119910 = ⟨120577 ℎ⟩ isin 119881 we have P-almost surely
119884119905and120591
= ⟨120577 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120577 120590119895
(119883119904)⟩ 119889120573
119895
119904
= ⟨120577 ℎ⟩ + int
119905and120591
0
120572120601120577
(⟨120577 119883119904⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(⟨120577 119883
119904⟩) 119889120573
119895
119904
= 119910 + int
119905and120591
0
120572120601120577
(119884119904) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(204)
showing that 119884 is a local strong solution to (192) with initialcondition 119910 By Itorsquos formula (Theorem 26) we obtain P-almost surely
119883119905and120591
= 120601 (119884119905and120591
)
= ℎ + int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904)))119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(205)
Now let 120585 isin D(119860lowast
) be arbitrary Then we have P-almostsurely
⟨120585 119883119905and120591
⟩
= ⟨120585 ℎ⟩
+ int
119905and120591
0
⟨120585119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 119863120601 (119884119904) 120590
119895
120601120577(119884
119904)⟩119889120573
119895
119904 119905 ge 0
(206)
On the other hand since 119883 is a local weak solution to (176)with initial condition ℎ and lifetime 120591 we have P-almostsurely for all 119905 ge 0 the identity
⟨120585 119883119905and120591
⟩ = ⟨120585 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904)⟩ 119889120573
119895
119904
(207)
Combining (206) and (207) yields up to indistinguishability
119861 + 119872 = 0 (208)
where the processes 119861 and 119872 are defined as
119861119905= int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩) 119889119904
119905 ge 0
119872119905= sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩ 119889120573
119895
119904
119905 ge 0
(209)
The process 119861 + 119872 is a continuous semimartingale withcanonical decomposition (208) Since the canonical decom-position of a continuous semimartingale is unique up toindistinguishability we deduce that 119861 = 119872 = 0 up toindistinguishability Using the Ito isometry (39) we obtain P-almost surely
int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩
+ ⟨120585 120572 (119883119904) minus 119863120601 (119884
showing that119883 is a local strong solution to (44) with lifetime120591
(3)rArr(1) This implication is a direct consequence ofProposition 29
The results from this section are closely related to the exis-tence of finite dimensional realizations that is the existenceof invariantmanifolds for each starting point ℎ
0 andwe point
out the papers [17ndash22] regarding this topic Furthermore wemention thatTheorem 68 has been extended in [23] to SPDEswith jumps
Acknowledgments
The author is grateful to Daniel Gaigall Georg GrafendorferFlorian Modler and Thomas Salfeld for the valuable com-ments and discussions
References
[1] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions CambridgeUniversity Press CambridgeUK 1992
[2] C Prevot and M Rockner A Concise Course on StochasticPartial Differential Equations Springer Berlin Germany 2007
[3] L Gawarecki and V Mandrekar Stochastic Differential Equa-tions in Infinite Dimensions with Applications to StochasticPartial Differential Equations Springer Berlin Germany 2011
[4] S Peszat and J Zabczyk Stochastic Partial Differential Equationswith Levy Noise Cambridge University Press Cambridge UK2007
[5] S Albeverio V Mandrekar and B Rudiger ldquoExistence of mildsolutions for stochastic differential equations and semilinearequations with non-Gaussian Levy noiserdquo Stochastic Processesand Their Applications vol 119 no 3 pp 835ndash863 2009
[6] D Filipovic S Tappe and J Teichmann ldquoJump-diffusions inHilbert spaces existence stability and numericsrdquo Stochasticsvol 82 no 5 pp 475ndash520 2010
[7] C Marinelli C Prevot and M Rockner ldquoRegular dependenceon initial data for stochastic evolution equations with multi-plicative Poisson noiserdquo Journal of Functional Analysis vol 258no 2 pp 616ndash649 2010
[8] S Tappe ldquoSome refinements of existence results for SPDEsdriven by Wiener processes and Poisson random measuresrdquoInternational Journal of Stochastic Analysis vol 2012 Article ID236327 24 pages 2012
[9] T Nakayama ldquoSupport theorem for mild solutions of SDErsquos inHilbert spacesrdquo Journal of Mathematical Sciences the Universityof Tokyo vol 11 no 3 pp 245ndash311 2004
[10] T Nakayama ldquoViability theorem for SPDErsquos including HJMframeworkrdquo Journal of Mathematical Sciences the University ofTokyo vol 11 no 3 pp 313ndash324 2004
[11] W Rudin Functional Analysis McGraw-Hill New York NYUSA 2nd edition 1991
[12] DWerner Funktionalanalysis Springer Berlin Germany 2007[13] C D Aliprantis and K C Border Infinite Dimensional Analysis
Springer Berlin Germany 3rd edition 2006[14] S N Ethier and T G KurtzMarkov Processes Characterization
andConvergenceWiley Series in Probability and Statistics JohnWiley amp Sons New Jersey NJ USA 1986
[15] V Mandrekar and B Rudiger ldquoExistence and uniqueness ofpath wise solutions for stochastic integral equations driven byLevy noise on separable Banach spacesrdquo Stochastics vol 78 no4 pp 189ndash212 2006
[16] D Filipovic Consistency Problems for Heath-Jarrow-MortonInterest Rate Models Springer Berlin Germany 2001
[17] T Bjork and L Svensson ldquoOn the existence of finite-dimensional realizations for nonlinear forward rate modelsrdquoMathematical Finance vol 11 no 2 pp 205ndash243 2001
[18] T Bjork and C Landen ldquoOn the construction of finite dimen-sional realizations for nonlinear forward rate modelsrdquo Financeand Stochastics vol 6 no 3 pp 303ndash331 2002
[19] D Filipovic and J Teichmann ldquoExistence of invariantmanifoldsfor stochastic equations in infinite dimensionrdquo Journal ofFunctional Analysis vol 197 no 2 pp 398ndash432 2003
[20] D Filipovic and J Teichmann ldquoOn the geometry of the termstructure of interest ratesrdquo Proceedings of The Royal Society ofLondon A vol 460 no 2041 pp 129ndash167 2004
[21] S Tappe ldquoAn alternative approach on the existence of affinerealizations for HJM term structure modelsrdquo Proceedings of
International Journal of Stochastic Analysis 25
TheRoyal Society of LondonA vol 466 no 2122 pp 3033ndash30602010
[22] S Tappe ldquoExistence of affine realizations for Levy term struc-ture modelsrdquo Proceedings of The Royal Society of London A vol468 no 2147 pp 3685ndash3704 2012
[23] D Filipovic S Tappe and J Teichmann ldquoInvariant manifoldswith boundary for jump-diffusionsrdquo httparxivorgabs12021076
and hence by the Cauchy-Schwarz inequality the Ito isom-etry (39) the growth estimate (9) from Lemma 10 and thelocal Lipschitz conditions (113) we obtain
Using Gronwallrsquos inequality (see Lemma 47) we deduce that119891 equiv 0 Thus by the continuity of the sample paths of 119883 and119884 we obtain
P(⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ) = 1 forall119899 isin N (120)
and hence by the continuity of the probability measureP weconclude that
P(⋂
119905ge0
119883119905and120591
1Γ= 119884
119905and1205911Γ)
= P(⋂
119899isinN
⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ)
= lim119899rarrinfin
P(⋂
119905ge0
119883119905and120591119899
1Γ= 119884
119905and120591119899
1Γ) = 1
(121)
which completes the proof
The local Lipschitz conditions (113) are in general notsufficient in order to ensure the existence of mild solutions tothe SPDE (44) Now we will prove that the existence of mildsolutions follows from global Lipschitz and linear growthconditions on 120572 and 120590 For this we recall an auxiliary resultwhich extends the Ito isometry (39)
Lemma 49 Let 119879 ge 0 be arbitrary and let 119883 = (119883119905)119905isin[0119879]
for all 119905 ge 0 and all ℎ isin 119867 Then for every F0-measurable
random variable ℎ0
Ω rarr 119867 there exists a (up toindistinguishability) unique mild solution 119883 to (44)
Proof The uniqueness of mild solutions to (44) is a directconsequence of Theorem 48 and hence we may concentrateon the existence proof which we divide into the followingseveral steps
Step 1 First we suppose that the initial condition ℎ0satisfies
E[ℎ02119901
] lt infin for some 119901 gt 1 Let 119879 ge 0 be arbitrary Wedefine the Banach space
1198712119901
119879(119867) = 119871
2119901
(Ω times [0 119879] P119879P otimes 119889119905119867) (129)
and prove that the variation of constants equation
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(130)
has a unique solution in the space 1198712119901
119879(119867) This is done in the
following three steps
Step 11 For 119883 isin 1198712119901
119879(119867) we define the process Φ119883 by
(Φ119883)119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(131)
Then the process Φ119883 is well defined Indeed by the growthestimate (9) the linear growth condition (127) and Holderrsquosinequality we have
Consequently there exists an index 119899 isin N such that Φ119899 is
a contraction and hence according to the extension of theBanach fixed point theorem (see Corollary 46) the mappingΦ has a unique fixed point 119883 isin 119871
2119901
119879(119867) This fixed point 119883 is
a solution to the variation of constants equation (130) Since119879 ge 0 was arbitrary there exists a process 119883 which is a solu-tion of the variation of constants equation
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
119905 ge 0
(146)
Step 14 In order to prove that 119883 is a mild solution to (44)it remains to show that 119883 has a continuous version ByLemma 12 the process
119905 997891997888rarr 119878119905ℎ0 119905 ge 0 (147)
is continuous and by Proposition 41 the process
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 119905 ge 0 (148)
is continuous too Moreover for every 119879 ge 0 we have by thelinear growth condition (128) Holderrsquos inequality and since119883 isin 119871
Thus by Proposition 43 the stochastic convolution 120590 ⋆ 119882
given by
(120590 ⋆ 119882)119905= int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0 (150)
has a continuous version and consequently the process119883hasa continuous version too This continuous version is a mildsolution to (44)
Step 2 Now let ℎ0 Ω rarr 119867 be an arbitraryF
0-measurable
random variable We define the sequence (ℎ119899)119899isinN of F
0-
measurable random variables as
ℎ119899
0= ℎ
01ℎ0le119899
119899 isin N (151)
Let 119899 isin N be arbitrary Then as ℎ119899
0is bounded we have
E[ℎ119899
02119901
] lt infin for all 119901 gt 1 By Step 1 the SPDE
119889119883119899
119905= (119860119883
119899
119905+ 120572 (119905 119883
119899
119905)) 119889119905 + 120590 (119905 119883
119899
119905) 119889119882
119905
119883119899
0= ℎ
119899
0
(152)
18 International Journal of Stochastic Analysis
has amild solution119883119899We define the sequence (Ω
119899)119899isinN sub F
0
as
Ω119899=
1003817100381710038171003817ℎ01003817100381710038171003817 le 119899 119899 isin N (153)
Then we have Ω119899sub Ω
119898for 119899 le 119898 Ω = ⋃
119899isinN Ω119899 and
Ω119899sub ℎ
119899
0= ℎ
119898
0 sub ℎ
119899
0= ℎ
0 forall119899 le 119898 (154)
Thus byTheorem 48 we have (up to indistinguishability)
119883119899
1Ω119899
= 119883119898
1Ω119899
forall119899 le 119898 (155)
Consequently the process
119883 = lim119899rarrinfin
119883119899
1Ω119899
(156)
is a well-defined continuous and adapted process and wehave
119883119899
1Ω119899
= 119883119898
1Ω119899
= 1198831Ω119899
forall119899 le 119898 (157)
Furthermore we obtain P-almost surely
119883119905= lim
119899rarrinfin
119883119899
1199051Ω119899
= lim119899rarrinfin
1Ω119899
(119878119905ℎ119899
0+ int
119905
0
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+int
119905
0
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ119899
0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= lim119899rarrinfin
1Ω119899
(119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= 119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(158)
proving that 119883 is a mild solution to (44)
Remark 51 For the proof of Theorem 50 we have usedCorollary 46 which is a slight extension of the Banach fixedpoint theorem Such an idea has been applied for example in[15]
Remark 52 A recent method for proving existence anduniqueness of mild solutions to the SPDE (44) is the methodof the moving frame presented in [6] see also [8] It allowsto reduce SPDE problems to the study of SDEs in infinitedimension In order to apply this method we need that thesemigroup (119878
119905)119905ge0
is a semigroup of pseudocontractions
We close this section with a consequence about theexistence of weak solutions
Corollary 53 Suppose that conditions (125)ndash(128) are ful-filled Let ℎ
0 Ω rarr 119867 be aF
0-measurable random variable
such that E[ℎ02119901
] lt infin for some 119901 gt 1 Then there exists a(up to indistinguishability) unique weak solution 119883 to (44)
Proof According to Proposition 35 every weak solution119883 to(44) is also a mild solution to (44)Therefore the uniquenessof weak solutions to (44) is a consequence of Theorem 48
It remains to prove the existence of a weak solution to(44) Let 119879 ge 0 be arbitrary By Theorem 50 and its proofthere exists a mild solution119883 isin 119871
2119901
119879(119867) to (44) By the linear
growth condition (128) and Holderrsquos inequality we obtain
showing that condition (72) is fulfilled Thus byProposition 37 the process 119883 is also a weak solution to(44)
8 Invariant Manifolds for WeakSolutions to SPDEs
In this section we deal with invariant manifolds for time-homogeneous SPDEs of the type (44) This topic arises fromthe natural desire to express the solutions of the SPDE (44)which generally live in the infinite dimensional Hilbert space119867 by means of a finite dimensional state process and thusto ensure larger analytical tractability Our goal is to findconditions on the generator 119860 and the coefficients 120572 120590 suchthat for every starting point of a finite dimensional submani-fold the solution process stays on the submanifold
We start with the required preliminaries about finitedimensional submanifolds inHilbert spaces In the sequel let119867 be a separable Hilbert space
Definition 54 Let 119898 119896 isin N be positive integers A subsetM sub 119867 is called an 119898-dimensional 119862119896-submanifold of 119867
International Journal of Stochastic Analysis 19
if for every ℎ isin M there exist an open neighborhood 119880 sub 119867
of ℎ an open set 119881 sub R119898 and a mapping 120601 isin 1198622
(119881119867) suchthat
(1) the mapping 120601 119881 rarr 119880 cap M is a homeomorphism(2) for all 119910 isin 119881 the mapping 119863120601(119910) is injective
The mapping 120601 is called a parametrization ofM around ℎ
In what follows let M be an 119898-dimensional 119862119896-
submanifold of 119867
Lemma 55 Let 120601119894
119881119894
rarr 119880119894cap M 119894 = 1 2 be two
parametrizations with119882= 1198801cap119880
2capM = 0Then themapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (160)
is a 119862119896-diffeomorphism
Proof See [16 Lemma 611]
Corollary 56 Let ℎ isin M be arbitrary and let 120601119894
119881119894
rarr
119880119894capM 119894 = 1 2 be two parametrizations ofM around ℎ Then
one has
1198631206011(119910
1) (R
119898
) = 1198631206012(119910
2) (R
119898
) (161)
where 119910119894= 120601
minus1
119894(ℎ) for 119894 = 1 2
Proof Since1198801and119880
2are open neighborhoods of ℎ we have
119882= 1198801cap 119880
2cap M = 0 Thus by Lemma 55 the mapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (162)
is a 119862119896-diffeomorphism Using the chain rule we obtain
1198631206012(119910
2) (R
119898
)
= 119863 (1206011∘ (120601
minus1
1∘ 120601
2)) (119910
2) (R
119898
)
= 1198631206011(119910
1)119863 (120601
minus1
1∘ 120601
2) (119910
2) (R
119898
)
sub 1198631206011(119910
1) (R
119898
)
(163)
and analogously we prove that 1198631206011(119910
1)(R119898
) sub
1198631206012(119910
2)(R119898
)
Definition 57 Let ℎ isin M be arbitraryThe tangent space ofMto ℎ is the subspace
119879ℎM = 119863120601 (119910) (R
119898
) (164)
where 119910 = 120601minus1
(ℎ) and 120601 119881 rarr 119880 cap M denotes aparametrization ofM around ℎ
Remark 58 Note that according to Corollary 56 theDefinition 57 of the tangent space 119879
ℎM does not depend on
the choice of the parametrization 120601 119881 rarr 119880 cap M
Proposition 59 Let ℎ isin M be arbitrary and let 120601 119881 rarr
119880 cap M be a parametrization ofM around ℎ Then there existan open set 119881
0sub 119881 an open neighborhood 119880
0sub 119880 of ℎ and
a mapping 120601 isin 119862119896
119887(R119898
119867) with 120601|1198810
= 120601|1198810
such that 120601|1198810
1198810
rarr 1198800cap M is a parametrization ofM around ℎ too
Proof See [16 Remark 611]
Remark 60 By Proposition 59 we may assume that anyparametrization 120601 119881 rarr 119880 cap M has an extension 120601 isin
119862119896
119887(R119898
119867)
Proposition 61 Let 119863 sub 119867 be a dense subset For every ℎ0isin
M there exist 1205771 120577
119898isin 119863 and a parametrization 120601 119881 rarr
119880 cap M around ℎ0such that
120601 (⟨120577 ℎ⟩) = ℎ forallℎ isin 119880 cap M (165)
where one uses the notation ⟨120577 ℎ⟩ = (⟨1205771 ℎ⟩ ⟨120577
119898 ℎ⟩) isin
R119898
Proof See [16 Proposition 612]
Proposition 62 Let 120601 119881 rarr 119880 cap M be a parametrizationas in Proposition 61 Then the following statements are true
(1) The elements 1205771 120577
119898are linearly independent in 119867
(2) For every ℎ isin 119880 cap M one has the direct sum decom-position
Therefore by Proposition 64 for every 1199100isin R119898 there exists
a (up to indistinguishability) unique weak solution to (192)which according to Proposition 39 is also a strong solutionto (192) The uniqueness of strong solutions to (192) is a con-sequence of Proposition 39 andTheorem 48
Now we are ready to formulate and prove ourmain resultof this section
Theorem 68 The following statements are equivalent
(1) The submanifoldM is locally invariant for (176)(2) One has
M sub D (119860) (195)
120590119895
(ℎ) isin 119879ℎM forallℎ isin M 119886119897119897 119895 isin N (196)
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) isin 119879ℎM forallℎ isin M (197)
(3) The operator 119860 is continuous onM and for each ℎ0isin
M there exists a local strong solution 119883 to (176) withsome lifetime 120591 gt 0 such that
119883119905and120591
isin M forall119905 ge 0 P-119886119897119898119900119904119905 119904119906119903119890119897119910 (198)
Proof (1)rArr(2) Let ℎ isin M be arbitrary By Proposition 61and Remark 60 there exist elements 120577
1 120577
119898isin D(119860
lowast
) anda parametrization 120601 119881 rarr 119880 cap M around ℎ such that theinverse 120601
minus1
119880 cap M rarr 119881 is given by 120601minus1
= ⟨120577 ∙⟩ and 120601
has an extension 120601 isin 1198622
119887(R119898
119867) Since the submanifoldM islocally invariant for (176) there exists a local weak solution119883
to (176) with initial condition ℎ and some lifetime 984858 gt 0 suchthat
119883119905and984858
isin M forall119905 ge 0 P-almost surely (199)
Since119880 is an open neighborhood of ℎ there exists 120598 gt 0 suchthat 119861
120598(ℎ) sub 119880 where 119861
120598(ℎ) denotes the open ball
119861120598(ℎ) = 119892 isin 119867
1003817100381710038171003817119892 minus ℎ1003817100381710038171003817 lt 120598 (200)
We define the stopping time
120591 = 984858 and inf 119905 ge 0 119883119905notin 119861
120598(ℎ) (201)
Since the process119883 has continuous sample paths and satisfies1198830= ℎ we have 120591 gt 0 and P-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (202)
22 International Journal of Stochastic Analysis
Defining the R119898-valued process 119884 = ⟨120577 119883⟩ we have P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (203)
Moreover since 119883 is a weak solution to (176) with initialcondition ℎ setting 119910 = ⟨120577 ℎ⟩ isin 119881 we have P-almost surely
119884119905and120591
= ⟨120577 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120577 120590119895
(119883119904)⟩ 119889120573
119895
119904
= ⟨120577 ℎ⟩ + int
119905and120591
0
120572120601120577
(⟨120577 119883119904⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(⟨120577 119883
119904⟩) 119889120573
119895
119904
= 119910 + int
119905and120591
0
120572120601120577
(119884119904) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(204)
showing that 119884 is a local strong solution to (192) with initialcondition 119910 By Itorsquos formula (Theorem 26) we obtain P-almost surely
119883119905and120591
= 120601 (119884119905and120591
)
= ℎ + int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904)))119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(205)
Now let 120585 isin D(119860lowast
) be arbitrary Then we have P-almostsurely
⟨120585 119883119905and120591
⟩
= ⟨120585 ℎ⟩
+ int
119905and120591
0
⟨120585119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 119863120601 (119884119904) 120590
119895
120601120577(119884
119904)⟩119889120573
119895
119904 119905 ge 0
(206)
On the other hand since 119883 is a local weak solution to (176)with initial condition ℎ and lifetime 120591 we have P-almostsurely for all 119905 ge 0 the identity
⟨120585 119883119905and120591
⟩ = ⟨120585 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904)⟩ 119889120573
119895
119904
(207)
Combining (206) and (207) yields up to indistinguishability
119861 + 119872 = 0 (208)
where the processes 119861 and 119872 are defined as
119861119905= int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩) 119889119904
119905 ge 0
119872119905= sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩ 119889120573
119895
119904
119905 ge 0
(209)
The process 119861 + 119872 is a continuous semimartingale withcanonical decomposition (208) Since the canonical decom-position of a continuous semimartingale is unique up toindistinguishability we deduce that 119861 = 119872 = 0 up toindistinguishability Using the Ito isometry (39) we obtain P-almost surely
int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩
+ ⟨120585 120572 (119883119904) minus 119863120601 (119884
showing that119883 is a local strong solution to (44) with lifetime120591
(3)rArr(1) This implication is a direct consequence ofProposition 29
The results from this section are closely related to the exis-tence of finite dimensional realizations that is the existenceof invariantmanifolds for each starting point ℎ
0 andwe point
out the papers [17ndash22] regarding this topic Furthermore wemention thatTheorem 68 has been extended in [23] to SPDEswith jumps
Acknowledgments
The author is grateful to Daniel Gaigall Georg GrafendorferFlorian Modler and Thomas Salfeld for the valuable com-ments and discussions
References
[1] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions CambridgeUniversity Press CambridgeUK 1992
[2] C Prevot and M Rockner A Concise Course on StochasticPartial Differential Equations Springer Berlin Germany 2007
[3] L Gawarecki and V Mandrekar Stochastic Differential Equa-tions in Infinite Dimensions with Applications to StochasticPartial Differential Equations Springer Berlin Germany 2011
[4] S Peszat and J Zabczyk Stochastic Partial Differential Equationswith Levy Noise Cambridge University Press Cambridge UK2007
[5] S Albeverio V Mandrekar and B Rudiger ldquoExistence of mildsolutions for stochastic differential equations and semilinearequations with non-Gaussian Levy noiserdquo Stochastic Processesand Their Applications vol 119 no 3 pp 835ndash863 2009
[6] D Filipovic S Tappe and J Teichmann ldquoJump-diffusions inHilbert spaces existence stability and numericsrdquo Stochasticsvol 82 no 5 pp 475ndash520 2010
[7] C Marinelli C Prevot and M Rockner ldquoRegular dependenceon initial data for stochastic evolution equations with multi-plicative Poisson noiserdquo Journal of Functional Analysis vol 258no 2 pp 616ndash649 2010
[8] S Tappe ldquoSome refinements of existence results for SPDEsdriven by Wiener processes and Poisson random measuresrdquoInternational Journal of Stochastic Analysis vol 2012 Article ID236327 24 pages 2012
[9] T Nakayama ldquoSupport theorem for mild solutions of SDErsquos inHilbert spacesrdquo Journal of Mathematical Sciences the Universityof Tokyo vol 11 no 3 pp 245ndash311 2004
[10] T Nakayama ldquoViability theorem for SPDErsquos including HJMframeworkrdquo Journal of Mathematical Sciences the University ofTokyo vol 11 no 3 pp 313ndash324 2004
[11] W Rudin Functional Analysis McGraw-Hill New York NYUSA 2nd edition 1991
[12] DWerner Funktionalanalysis Springer Berlin Germany 2007[13] C D Aliprantis and K C Border Infinite Dimensional Analysis
Springer Berlin Germany 3rd edition 2006[14] S N Ethier and T G KurtzMarkov Processes Characterization
andConvergenceWiley Series in Probability and Statistics JohnWiley amp Sons New Jersey NJ USA 1986
[15] V Mandrekar and B Rudiger ldquoExistence and uniqueness ofpath wise solutions for stochastic integral equations driven byLevy noise on separable Banach spacesrdquo Stochastics vol 78 no4 pp 189ndash212 2006
[16] D Filipovic Consistency Problems for Heath-Jarrow-MortonInterest Rate Models Springer Berlin Germany 2001
[17] T Bjork and L Svensson ldquoOn the existence of finite-dimensional realizations for nonlinear forward rate modelsrdquoMathematical Finance vol 11 no 2 pp 205ndash243 2001
[18] T Bjork and C Landen ldquoOn the construction of finite dimen-sional realizations for nonlinear forward rate modelsrdquo Financeand Stochastics vol 6 no 3 pp 303ndash331 2002
[19] D Filipovic and J Teichmann ldquoExistence of invariantmanifoldsfor stochastic equations in infinite dimensionrdquo Journal ofFunctional Analysis vol 197 no 2 pp 398ndash432 2003
[20] D Filipovic and J Teichmann ldquoOn the geometry of the termstructure of interest ratesrdquo Proceedings of The Royal Society ofLondon A vol 460 no 2041 pp 129ndash167 2004
[21] S Tappe ldquoAn alternative approach on the existence of affinerealizations for HJM term structure modelsrdquo Proceedings of
International Journal of Stochastic Analysis 25
TheRoyal Society of LondonA vol 466 no 2122 pp 3033ndash30602010
[22] S Tappe ldquoExistence of affine realizations for Levy term struc-ture modelsrdquo Proceedings of The Royal Society of London A vol468 no 2147 pp 3685ndash3704 2012
[23] D Filipovic S Tappe and J Teichmann ldquoInvariant manifoldswith boundary for jump-diffusionsrdquo httparxivorgabs12021076
for all 119905 ge 0 and all ℎ isin 119867 Then for every F0-measurable
random variable ℎ0
Ω rarr 119867 there exists a (up toindistinguishability) unique mild solution 119883 to (44)
Proof The uniqueness of mild solutions to (44) is a directconsequence of Theorem 48 and hence we may concentrateon the existence proof which we divide into the followingseveral steps
Step 1 First we suppose that the initial condition ℎ0satisfies
E[ℎ02119901
] lt infin for some 119901 gt 1 Let 119879 ge 0 be arbitrary Wedefine the Banach space
1198712119901
119879(119867) = 119871
2119901
(Ω times [0 119879] P119879P otimes 119889119905119867) (129)
and prove that the variation of constants equation
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(130)
has a unique solution in the space 1198712119901
119879(119867) This is done in the
following three steps
Step 11 For 119883 isin 1198712119901
119879(119867) we define the process Φ119883 by
(Φ119883)119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 isin [0 119879]
(131)
Then the process Φ119883 is well defined Indeed by the growthestimate (9) the linear growth condition (127) and Holderrsquosinequality we have
Consequently there exists an index 119899 isin N such that Φ119899 is
a contraction and hence according to the extension of theBanach fixed point theorem (see Corollary 46) the mappingΦ has a unique fixed point 119883 isin 119871
2119901
119879(119867) This fixed point 119883 is
a solution to the variation of constants equation (130) Since119879 ge 0 was arbitrary there exists a process 119883 which is a solu-tion of the variation of constants equation
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
119905 ge 0
(146)
Step 14 In order to prove that 119883 is a mild solution to (44)it remains to show that 119883 has a continuous version ByLemma 12 the process
119905 997891997888rarr 119878119905ℎ0 119905 ge 0 (147)
is continuous and by Proposition 41 the process
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 119905 ge 0 (148)
is continuous too Moreover for every 119879 ge 0 we have by thelinear growth condition (128) Holderrsquos inequality and since119883 isin 119871
Thus by Proposition 43 the stochastic convolution 120590 ⋆ 119882
given by
(120590 ⋆ 119882)119905= int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0 (150)
has a continuous version and consequently the process119883hasa continuous version too This continuous version is a mildsolution to (44)
Step 2 Now let ℎ0 Ω rarr 119867 be an arbitraryF
0-measurable
random variable We define the sequence (ℎ119899)119899isinN of F
0-
measurable random variables as
ℎ119899
0= ℎ
01ℎ0le119899
119899 isin N (151)
Let 119899 isin N be arbitrary Then as ℎ119899
0is bounded we have
E[ℎ119899
02119901
] lt infin for all 119901 gt 1 By Step 1 the SPDE
119889119883119899
119905= (119860119883
119899
119905+ 120572 (119905 119883
119899
119905)) 119889119905 + 120590 (119905 119883
119899
119905) 119889119882
119905
119883119899
0= ℎ
119899
0
(152)
18 International Journal of Stochastic Analysis
has amild solution119883119899We define the sequence (Ω
119899)119899isinN sub F
0
as
Ω119899=
1003817100381710038171003817ℎ01003817100381710038171003817 le 119899 119899 isin N (153)
Then we have Ω119899sub Ω
119898for 119899 le 119898 Ω = ⋃
119899isinN Ω119899 and
Ω119899sub ℎ
119899
0= ℎ
119898
0 sub ℎ
119899
0= ℎ
0 forall119899 le 119898 (154)
Thus byTheorem 48 we have (up to indistinguishability)
119883119899
1Ω119899
= 119883119898
1Ω119899
forall119899 le 119898 (155)
Consequently the process
119883 = lim119899rarrinfin
119883119899
1Ω119899
(156)
is a well-defined continuous and adapted process and wehave
119883119899
1Ω119899
= 119883119898
1Ω119899
= 1198831Ω119899
forall119899 le 119898 (157)
Furthermore we obtain P-almost surely
119883119905= lim
119899rarrinfin
119883119899
1199051Ω119899
= lim119899rarrinfin
1Ω119899
(119878119905ℎ119899
0+ int
119905
0
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+int
119905
0
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ119899
0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= lim119899rarrinfin
1Ω119899
(119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= 119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(158)
proving that 119883 is a mild solution to (44)
Remark 51 For the proof of Theorem 50 we have usedCorollary 46 which is a slight extension of the Banach fixedpoint theorem Such an idea has been applied for example in[15]
Remark 52 A recent method for proving existence anduniqueness of mild solutions to the SPDE (44) is the methodof the moving frame presented in [6] see also [8] It allowsto reduce SPDE problems to the study of SDEs in infinitedimension In order to apply this method we need that thesemigroup (119878
119905)119905ge0
is a semigroup of pseudocontractions
We close this section with a consequence about theexistence of weak solutions
Corollary 53 Suppose that conditions (125)ndash(128) are ful-filled Let ℎ
0 Ω rarr 119867 be aF
0-measurable random variable
such that E[ℎ02119901
] lt infin for some 119901 gt 1 Then there exists a(up to indistinguishability) unique weak solution 119883 to (44)
Proof According to Proposition 35 every weak solution119883 to(44) is also a mild solution to (44)Therefore the uniquenessof weak solutions to (44) is a consequence of Theorem 48
It remains to prove the existence of a weak solution to(44) Let 119879 ge 0 be arbitrary By Theorem 50 and its proofthere exists a mild solution119883 isin 119871
2119901
119879(119867) to (44) By the linear
growth condition (128) and Holderrsquos inequality we obtain
showing that condition (72) is fulfilled Thus byProposition 37 the process 119883 is also a weak solution to(44)
8 Invariant Manifolds for WeakSolutions to SPDEs
In this section we deal with invariant manifolds for time-homogeneous SPDEs of the type (44) This topic arises fromthe natural desire to express the solutions of the SPDE (44)which generally live in the infinite dimensional Hilbert space119867 by means of a finite dimensional state process and thusto ensure larger analytical tractability Our goal is to findconditions on the generator 119860 and the coefficients 120572 120590 suchthat for every starting point of a finite dimensional submani-fold the solution process stays on the submanifold
We start with the required preliminaries about finitedimensional submanifolds inHilbert spaces In the sequel let119867 be a separable Hilbert space
Definition 54 Let 119898 119896 isin N be positive integers A subsetM sub 119867 is called an 119898-dimensional 119862119896-submanifold of 119867
International Journal of Stochastic Analysis 19
if for every ℎ isin M there exist an open neighborhood 119880 sub 119867
of ℎ an open set 119881 sub R119898 and a mapping 120601 isin 1198622
(119881119867) suchthat
(1) the mapping 120601 119881 rarr 119880 cap M is a homeomorphism(2) for all 119910 isin 119881 the mapping 119863120601(119910) is injective
The mapping 120601 is called a parametrization ofM around ℎ
In what follows let M be an 119898-dimensional 119862119896-
submanifold of 119867
Lemma 55 Let 120601119894
119881119894
rarr 119880119894cap M 119894 = 1 2 be two
parametrizations with119882= 1198801cap119880
2capM = 0Then themapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (160)
is a 119862119896-diffeomorphism
Proof See [16 Lemma 611]
Corollary 56 Let ℎ isin M be arbitrary and let 120601119894
119881119894
rarr
119880119894capM 119894 = 1 2 be two parametrizations ofM around ℎ Then
one has
1198631206011(119910
1) (R
119898
) = 1198631206012(119910
2) (R
119898
) (161)
where 119910119894= 120601
minus1
119894(ℎ) for 119894 = 1 2
Proof Since1198801and119880
2are open neighborhoods of ℎ we have
119882= 1198801cap 119880
2cap M = 0 Thus by Lemma 55 the mapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (162)
is a 119862119896-diffeomorphism Using the chain rule we obtain
1198631206012(119910
2) (R
119898
)
= 119863 (1206011∘ (120601
minus1
1∘ 120601
2)) (119910
2) (R
119898
)
= 1198631206011(119910
1)119863 (120601
minus1
1∘ 120601
2) (119910
2) (R
119898
)
sub 1198631206011(119910
1) (R
119898
)
(163)
and analogously we prove that 1198631206011(119910
1)(R119898
) sub
1198631206012(119910
2)(R119898
)
Definition 57 Let ℎ isin M be arbitraryThe tangent space ofMto ℎ is the subspace
119879ℎM = 119863120601 (119910) (R
119898
) (164)
where 119910 = 120601minus1
(ℎ) and 120601 119881 rarr 119880 cap M denotes aparametrization ofM around ℎ
Remark 58 Note that according to Corollary 56 theDefinition 57 of the tangent space 119879
ℎM does not depend on
the choice of the parametrization 120601 119881 rarr 119880 cap M
Proposition 59 Let ℎ isin M be arbitrary and let 120601 119881 rarr
119880 cap M be a parametrization ofM around ℎ Then there existan open set 119881
0sub 119881 an open neighborhood 119880
0sub 119880 of ℎ and
a mapping 120601 isin 119862119896
119887(R119898
119867) with 120601|1198810
= 120601|1198810
such that 120601|1198810
1198810
rarr 1198800cap M is a parametrization ofM around ℎ too
Proof See [16 Remark 611]
Remark 60 By Proposition 59 we may assume that anyparametrization 120601 119881 rarr 119880 cap M has an extension 120601 isin
119862119896
119887(R119898
119867)
Proposition 61 Let 119863 sub 119867 be a dense subset For every ℎ0isin
M there exist 1205771 120577
119898isin 119863 and a parametrization 120601 119881 rarr
119880 cap M around ℎ0such that
120601 (⟨120577 ℎ⟩) = ℎ forallℎ isin 119880 cap M (165)
where one uses the notation ⟨120577 ℎ⟩ = (⟨1205771 ℎ⟩ ⟨120577
119898 ℎ⟩) isin
R119898
Proof See [16 Proposition 612]
Proposition 62 Let 120601 119881 rarr 119880 cap M be a parametrizationas in Proposition 61 Then the following statements are true
(1) The elements 1205771 120577
119898are linearly independent in 119867
(2) For every ℎ isin 119880 cap M one has the direct sum decom-position
Therefore by Proposition 64 for every 1199100isin R119898 there exists
a (up to indistinguishability) unique weak solution to (192)which according to Proposition 39 is also a strong solutionto (192) The uniqueness of strong solutions to (192) is a con-sequence of Proposition 39 andTheorem 48
Now we are ready to formulate and prove ourmain resultof this section
Theorem 68 The following statements are equivalent
(1) The submanifoldM is locally invariant for (176)(2) One has
M sub D (119860) (195)
120590119895
(ℎ) isin 119879ℎM forallℎ isin M 119886119897119897 119895 isin N (196)
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) isin 119879ℎM forallℎ isin M (197)
(3) The operator 119860 is continuous onM and for each ℎ0isin
M there exists a local strong solution 119883 to (176) withsome lifetime 120591 gt 0 such that
119883119905and120591
isin M forall119905 ge 0 P-119886119897119898119900119904119905 119904119906119903119890119897119910 (198)
Proof (1)rArr(2) Let ℎ isin M be arbitrary By Proposition 61and Remark 60 there exist elements 120577
1 120577
119898isin D(119860
lowast
) anda parametrization 120601 119881 rarr 119880 cap M around ℎ such that theinverse 120601
minus1
119880 cap M rarr 119881 is given by 120601minus1
= ⟨120577 ∙⟩ and 120601
has an extension 120601 isin 1198622
119887(R119898
119867) Since the submanifoldM islocally invariant for (176) there exists a local weak solution119883
to (176) with initial condition ℎ and some lifetime 984858 gt 0 suchthat
119883119905and984858
isin M forall119905 ge 0 P-almost surely (199)
Since119880 is an open neighborhood of ℎ there exists 120598 gt 0 suchthat 119861
120598(ℎ) sub 119880 where 119861
120598(ℎ) denotes the open ball
119861120598(ℎ) = 119892 isin 119867
1003817100381710038171003817119892 minus ℎ1003817100381710038171003817 lt 120598 (200)
We define the stopping time
120591 = 984858 and inf 119905 ge 0 119883119905notin 119861
120598(ℎ) (201)
Since the process119883 has continuous sample paths and satisfies1198830= ℎ we have 120591 gt 0 and P-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (202)
22 International Journal of Stochastic Analysis
Defining the R119898-valued process 119884 = ⟨120577 119883⟩ we have P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (203)
Moreover since 119883 is a weak solution to (176) with initialcondition ℎ setting 119910 = ⟨120577 ℎ⟩ isin 119881 we have P-almost surely
119884119905and120591
= ⟨120577 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120577 120590119895
(119883119904)⟩ 119889120573
119895
119904
= ⟨120577 ℎ⟩ + int
119905and120591
0
120572120601120577
(⟨120577 119883119904⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(⟨120577 119883
119904⟩) 119889120573
119895
119904
= 119910 + int
119905and120591
0
120572120601120577
(119884119904) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(204)
showing that 119884 is a local strong solution to (192) with initialcondition 119910 By Itorsquos formula (Theorem 26) we obtain P-almost surely
119883119905and120591
= 120601 (119884119905and120591
)
= ℎ + int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904)))119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(205)
Now let 120585 isin D(119860lowast
) be arbitrary Then we have P-almostsurely
⟨120585 119883119905and120591
⟩
= ⟨120585 ℎ⟩
+ int
119905and120591
0
⟨120585119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 119863120601 (119884119904) 120590
119895
120601120577(119884
119904)⟩119889120573
119895
119904 119905 ge 0
(206)
On the other hand since 119883 is a local weak solution to (176)with initial condition ℎ and lifetime 120591 we have P-almostsurely for all 119905 ge 0 the identity
⟨120585 119883119905and120591
⟩ = ⟨120585 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904)⟩ 119889120573
119895
119904
(207)
Combining (206) and (207) yields up to indistinguishability
119861 + 119872 = 0 (208)
where the processes 119861 and 119872 are defined as
119861119905= int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩) 119889119904
119905 ge 0
119872119905= sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩ 119889120573
119895
119904
119905 ge 0
(209)
The process 119861 + 119872 is a continuous semimartingale withcanonical decomposition (208) Since the canonical decom-position of a continuous semimartingale is unique up toindistinguishability we deduce that 119861 = 119872 = 0 up toindistinguishability Using the Ito isometry (39) we obtain P-almost surely
int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩
+ ⟨120585 120572 (119883119904) minus 119863120601 (119884
showing that119883 is a local strong solution to (44) with lifetime120591
(3)rArr(1) This implication is a direct consequence ofProposition 29
The results from this section are closely related to the exis-tence of finite dimensional realizations that is the existenceof invariantmanifolds for each starting point ℎ
0 andwe point
out the papers [17ndash22] regarding this topic Furthermore wemention thatTheorem 68 has been extended in [23] to SPDEswith jumps
Acknowledgments
The author is grateful to Daniel Gaigall Georg GrafendorferFlorian Modler and Thomas Salfeld for the valuable com-ments and discussions
References
[1] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions CambridgeUniversity Press CambridgeUK 1992
[2] C Prevot and M Rockner A Concise Course on StochasticPartial Differential Equations Springer Berlin Germany 2007
[3] L Gawarecki and V Mandrekar Stochastic Differential Equa-tions in Infinite Dimensions with Applications to StochasticPartial Differential Equations Springer Berlin Germany 2011
[4] S Peszat and J Zabczyk Stochastic Partial Differential Equationswith Levy Noise Cambridge University Press Cambridge UK2007
[5] S Albeverio V Mandrekar and B Rudiger ldquoExistence of mildsolutions for stochastic differential equations and semilinearequations with non-Gaussian Levy noiserdquo Stochastic Processesand Their Applications vol 119 no 3 pp 835ndash863 2009
[6] D Filipovic S Tappe and J Teichmann ldquoJump-diffusions inHilbert spaces existence stability and numericsrdquo Stochasticsvol 82 no 5 pp 475ndash520 2010
[7] C Marinelli C Prevot and M Rockner ldquoRegular dependenceon initial data for stochastic evolution equations with multi-plicative Poisson noiserdquo Journal of Functional Analysis vol 258no 2 pp 616ndash649 2010
[8] S Tappe ldquoSome refinements of existence results for SPDEsdriven by Wiener processes and Poisson random measuresrdquoInternational Journal of Stochastic Analysis vol 2012 Article ID236327 24 pages 2012
[9] T Nakayama ldquoSupport theorem for mild solutions of SDErsquos inHilbert spacesrdquo Journal of Mathematical Sciences the Universityof Tokyo vol 11 no 3 pp 245ndash311 2004
[10] T Nakayama ldquoViability theorem for SPDErsquos including HJMframeworkrdquo Journal of Mathematical Sciences the University ofTokyo vol 11 no 3 pp 313ndash324 2004
[11] W Rudin Functional Analysis McGraw-Hill New York NYUSA 2nd edition 1991
[12] DWerner Funktionalanalysis Springer Berlin Germany 2007[13] C D Aliprantis and K C Border Infinite Dimensional Analysis
Springer Berlin Germany 3rd edition 2006[14] S N Ethier and T G KurtzMarkov Processes Characterization
andConvergenceWiley Series in Probability and Statistics JohnWiley amp Sons New Jersey NJ USA 1986
[15] V Mandrekar and B Rudiger ldquoExistence and uniqueness ofpath wise solutions for stochastic integral equations driven byLevy noise on separable Banach spacesrdquo Stochastics vol 78 no4 pp 189ndash212 2006
[16] D Filipovic Consistency Problems for Heath-Jarrow-MortonInterest Rate Models Springer Berlin Germany 2001
[17] T Bjork and L Svensson ldquoOn the existence of finite-dimensional realizations for nonlinear forward rate modelsrdquoMathematical Finance vol 11 no 2 pp 205ndash243 2001
[18] T Bjork and C Landen ldquoOn the construction of finite dimen-sional realizations for nonlinear forward rate modelsrdquo Financeand Stochastics vol 6 no 3 pp 303ndash331 2002
[19] D Filipovic and J Teichmann ldquoExistence of invariantmanifoldsfor stochastic equations in infinite dimensionrdquo Journal ofFunctional Analysis vol 197 no 2 pp 398ndash432 2003
[20] D Filipovic and J Teichmann ldquoOn the geometry of the termstructure of interest ratesrdquo Proceedings of The Royal Society ofLondon A vol 460 no 2041 pp 129ndash167 2004
[21] S Tappe ldquoAn alternative approach on the existence of affinerealizations for HJM term structure modelsrdquo Proceedings of
International Journal of Stochastic Analysis 25
TheRoyal Society of LondonA vol 466 no 2122 pp 3033ndash30602010
[22] S Tappe ldquoExistence of affine realizations for Levy term struc-ture modelsrdquo Proceedings of The Royal Society of London A vol468 no 2147 pp 3685ndash3704 2012
[23] D Filipovic S Tappe and J Teichmann ldquoInvariant manifoldswith boundary for jump-diffusionsrdquo httparxivorgabs12021076
Consequently there exists an index 119899 isin N such that Φ119899 is
a contraction and hence according to the extension of theBanach fixed point theorem (see Corollary 46) the mappingΦ has a unique fixed point 119883 isin 119871
2119901
119879(119867) This fixed point 119883 is
a solution to the variation of constants equation (130) Since119879 ge 0 was arbitrary there exists a process 119883 which is a solu-tion of the variation of constants equation
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
119905 ge 0
(146)
Step 14 In order to prove that 119883 is a mild solution to (44)it remains to show that 119883 has a continuous version ByLemma 12 the process
119905 997891997888rarr 119878119905ℎ0 119905 ge 0 (147)
is continuous and by Proposition 41 the process
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 119905 ge 0 (148)
is continuous too Moreover for every 119879 ge 0 we have by thelinear growth condition (128) Holderrsquos inequality and since119883 isin 119871
Thus by Proposition 43 the stochastic convolution 120590 ⋆ 119882
given by
(120590 ⋆ 119882)119905= int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0 (150)
has a continuous version and consequently the process119883hasa continuous version too This continuous version is a mildsolution to (44)
Step 2 Now let ℎ0 Ω rarr 119867 be an arbitraryF
0-measurable
random variable We define the sequence (ℎ119899)119899isinN of F
0-
measurable random variables as
ℎ119899
0= ℎ
01ℎ0le119899
119899 isin N (151)
Let 119899 isin N be arbitrary Then as ℎ119899
0is bounded we have
E[ℎ119899
02119901
] lt infin for all 119901 gt 1 By Step 1 the SPDE
119889119883119899
119905= (119860119883
119899
119905+ 120572 (119905 119883
119899
119905)) 119889119905 + 120590 (119905 119883
119899
119905) 119889119882
119905
119883119899
0= ℎ
119899
0
(152)
18 International Journal of Stochastic Analysis
has amild solution119883119899We define the sequence (Ω
119899)119899isinN sub F
0
as
Ω119899=
1003817100381710038171003817ℎ01003817100381710038171003817 le 119899 119899 isin N (153)
Then we have Ω119899sub Ω
119898for 119899 le 119898 Ω = ⋃
119899isinN Ω119899 and
Ω119899sub ℎ
119899
0= ℎ
119898
0 sub ℎ
119899
0= ℎ
0 forall119899 le 119898 (154)
Thus byTheorem 48 we have (up to indistinguishability)
119883119899
1Ω119899
= 119883119898
1Ω119899
forall119899 le 119898 (155)
Consequently the process
119883 = lim119899rarrinfin
119883119899
1Ω119899
(156)
is a well-defined continuous and adapted process and wehave
119883119899
1Ω119899
= 119883119898
1Ω119899
= 1198831Ω119899
forall119899 le 119898 (157)
Furthermore we obtain P-almost surely
119883119905= lim
119899rarrinfin
119883119899
1199051Ω119899
= lim119899rarrinfin
1Ω119899
(119878119905ℎ119899
0+ int
119905
0
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+int
119905
0
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ119899
0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= lim119899rarrinfin
1Ω119899
(119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= 119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(158)
proving that 119883 is a mild solution to (44)
Remark 51 For the proof of Theorem 50 we have usedCorollary 46 which is a slight extension of the Banach fixedpoint theorem Such an idea has been applied for example in[15]
Remark 52 A recent method for proving existence anduniqueness of mild solutions to the SPDE (44) is the methodof the moving frame presented in [6] see also [8] It allowsto reduce SPDE problems to the study of SDEs in infinitedimension In order to apply this method we need that thesemigroup (119878
119905)119905ge0
is a semigroup of pseudocontractions
We close this section with a consequence about theexistence of weak solutions
Corollary 53 Suppose that conditions (125)ndash(128) are ful-filled Let ℎ
0 Ω rarr 119867 be aF
0-measurable random variable
such that E[ℎ02119901
] lt infin for some 119901 gt 1 Then there exists a(up to indistinguishability) unique weak solution 119883 to (44)
Proof According to Proposition 35 every weak solution119883 to(44) is also a mild solution to (44)Therefore the uniquenessof weak solutions to (44) is a consequence of Theorem 48
It remains to prove the existence of a weak solution to(44) Let 119879 ge 0 be arbitrary By Theorem 50 and its proofthere exists a mild solution119883 isin 119871
2119901
119879(119867) to (44) By the linear
growth condition (128) and Holderrsquos inequality we obtain
showing that condition (72) is fulfilled Thus byProposition 37 the process 119883 is also a weak solution to(44)
8 Invariant Manifolds for WeakSolutions to SPDEs
In this section we deal with invariant manifolds for time-homogeneous SPDEs of the type (44) This topic arises fromthe natural desire to express the solutions of the SPDE (44)which generally live in the infinite dimensional Hilbert space119867 by means of a finite dimensional state process and thusto ensure larger analytical tractability Our goal is to findconditions on the generator 119860 and the coefficients 120572 120590 suchthat for every starting point of a finite dimensional submani-fold the solution process stays on the submanifold
We start with the required preliminaries about finitedimensional submanifolds inHilbert spaces In the sequel let119867 be a separable Hilbert space
Definition 54 Let 119898 119896 isin N be positive integers A subsetM sub 119867 is called an 119898-dimensional 119862119896-submanifold of 119867
International Journal of Stochastic Analysis 19
if for every ℎ isin M there exist an open neighborhood 119880 sub 119867
of ℎ an open set 119881 sub R119898 and a mapping 120601 isin 1198622
(119881119867) suchthat
(1) the mapping 120601 119881 rarr 119880 cap M is a homeomorphism(2) for all 119910 isin 119881 the mapping 119863120601(119910) is injective
The mapping 120601 is called a parametrization ofM around ℎ
In what follows let M be an 119898-dimensional 119862119896-
submanifold of 119867
Lemma 55 Let 120601119894
119881119894
rarr 119880119894cap M 119894 = 1 2 be two
parametrizations with119882= 1198801cap119880
2capM = 0Then themapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (160)
is a 119862119896-diffeomorphism
Proof See [16 Lemma 611]
Corollary 56 Let ℎ isin M be arbitrary and let 120601119894
119881119894
rarr
119880119894capM 119894 = 1 2 be two parametrizations ofM around ℎ Then
one has
1198631206011(119910
1) (R
119898
) = 1198631206012(119910
2) (R
119898
) (161)
where 119910119894= 120601
minus1
119894(ℎ) for 119894 = 1 2
Proof Since1198801and119880
2are open neighborhoods of ℎ we have
119882= 1198801cap 119880
2cap M = 0 Thus by Lemma 55 the mapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (162)
is a 119862119896-diffeomorphism Using the chain rule we obtain
1198631206012(119910
2) (R
119898
)
= 119863 (1206011∘ (120601
minus1
1∘ 120601
2)) (119910
2) (R
119898
)
= 1198631206011(119910
1)119863 (120601
minus1
1∘ 120601
2) (119910
2) (R
119898
)
sub 1198631206011(119910
1) (R
119898
)
(163)
and analogously we prove that 1198631206011(119910
1)(R119898
) sub
1198631206012(119910
2)(R119898
)
Definition 57 Let ℎ isin M be arbitraryThe tangent space ofMto ℎ is the subspace
119879ℎM = 119863120601 (119910) (R
119898
) (164)
where 119910 = 120601minus1
(ℎ) and 120601 119881 rarr 119880 cap M denotes aparametrization ofM around ℎ
Remark 58 Note that according to Corollary 56 theDefinition 57 of the tangent space 119879
ℎM does not depend on
the choice of the parametrization 120601 119881 rarr 119880 cap M
Proposition 59 Let ℎ isin M be arbitrary and let 120601 119881 rarr
119880 cap M be a parametrization ofM around ℎ Then there existan open set 119881
0sub 119881 an open neighborhood 119880
0sub 119880 of ℎ and
a mapping 120601 isin 119862119896
119887(R119898
119867) with 120601|1198810
= 120601|1198810
such that 120601|1198810
1198810
rarr 1198800cap M is a parametrization ofM around ℎ too
Proof See [16 Remark 611]
Remark 60 By Proposition 59 we may assume that anyparametrization 120601 119881 rarr 119880 cap M has an extension 120601 isin
119862119896
119887(R119898
119867)
Proposition 61 Let 119863 sub 119867 be a dense subset For every ℎ0isin
M there exist 1205771 120577
119898isin 119863 and a parametrization 120601 119881 rarr
119880 cap M around ℎ0such that
120601 (⟨120577 ℎ⟩) = ℎ forallℎ isin 119880 cap M (165)
where one uses the notation ⟨120577 ℎ⟩ = (⟨1205771 ℎ⟩ ⟨120577
119898 ℎ⟩) isin
R119898
Proof See [16 Proposition 612]
Proposition 62 Let 120601 119881 rarr 119880 cap M be a parametrizationas in Proposition 61 Then the following statements are true
(1) The elements 1205771 120577
119898are linearly independent in 119867
(2) For every ℎ isin 119880 cap M one has the direct sum decom-position
Therefore by Proposition 64 for every 1199100isin R119898 there exists
a (up to indistinguishability) unique weak solution to (192)which according to Proposition 39 is also a strong solutionto (192) The uniqueness of strong solutions to (192) is a con-sequence of Proposition 39 andTheorem 48
Now we are ready to formulate and prove ourmain resultof this section
Theorem 68 The following statements are equivalent
(1) The submanifoldM is locally invariant for (176)(2) One has
M sub D (119860) (195)
120590119895
(ℎ) isin 119879ℎM forallℎ isin M 119886119897119897 119895 isin N (196)
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) isin 119879ℎM forallℎ isin M (197)
(3) The operator 119860 is continuous onM and for each ℎ0isin
M there exists a local strong solution 119883 to (176) withsome lifetime 120591 gt 0 such that
119883119905and120591
isin M forall119905 ge 0 P-119886119897119898119900119904119905 119904119906119903119890119897119910 (198)
Proof (1)rArr(2) Let ℎ isin M be arbitrary By Proposition 61and Remark 60 there exist elements 120577
1 120577
119898isin D(119860
lowast
) anda parametrization 120601 119881 rarr 119880 cap M around ℎ such that theinverse 120601
minus1
119880 cap M rarr 119881 is given by 120601minus1
= ⟨120577 ∙⟩ and 120601
has an extension 120601 isin 1198622
119887(R119898
119867) Since the submanifoldM islocally invariant for (176) there exists a local weak solution119883
to (176) with initial condition ℎ and some lifetime 984858 gt 0 suchthat
119883119905and984858
isin M forall119905 ge 0 P-almost surely (199)
Since119880 is an open neighborhood of ℎ there exists 120598 gt 0 suchthat 119861
120598(ℎ) sub 119880 where 119861
120598(ℎ) denotes the open ball
119861120598(ℎ) = 119892 isin 119867
1003817100381710038171003817119892 minus ℎ1003817100381710038171003817 lt 120598 (200)
We define the stopping time
120591 = 984858 and inf 119905 ge 0 119883119905notin 119861
120598(ℎ) (201)
Since the process119883 has continuous sample paths and satisfies1198830= ℎ we have 120591 gt 0 and P-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (202)
22 International Journal of Stochastic Analysis
Defining the R119898-valued process 119884 = ⟨120577 119883⟩ we have P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (203)
Moreover since 119883 is a weak solution to (176) with initialcondition ℎ setting 119910 = ⟨120577 ℎ⟩ isin 119881 we have P-almost surely
119884119905and120591
= ⟨120577 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120577 120590119895
(119883119904)⟩ 119889120573
119895
119904
= ⟨120577 ℎ⟩ + int
119905and120591
0
120572120601120577
(⟨120577 119883119904⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(⟨120577 119883
119904⟩) 119889120573
119895
119904
= 119910 + int
119905and120591
0
120572120601120577
(119884119904) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(204)
showing that 119884 is a local strong solution to (192) with initialcondition 119910 By Itorsquos formula (Theorem 26) we obtain P-almost surely
119883119905and120591
= 120601 (119884119905and120591
)
= ℎ + int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904)))119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(205)
Now let 120585 isin D(119860lowast
) be arbitrary Then we have P-almostsurely
⟨120585 119883119905and120591
⟩
= ⟨120585 ℎ⟩
+ int
119905and120591
0
⟨120585119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 119863120601 (119884119904) 120590
119895
120601120577(119884
119904)⟩119889120573
119895
119904 119905 ge 0
(206)
On the other hand since 119883 is a local weak solution to (176)with initial condition ℎ and lifetime 120591 we have P-almostsurely for all 119905 ge 0 the identity
⟨120585 119883119905and120591
⟩ = ⟨120585 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904)⟩ 119889120573
119895
119904
(207)
Combining (206) and (207) yields up to indistinguishability
119861 + 119872 = 0 (208)
where the processes 119861 and 119872 are defined as
119861119905= int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩) 119889119904
119905 ge 0
119872119905= sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩ 119889120573
119895
119904
119905 ge 0
(209)
The process 119861 + 119872 is a continuous semimartingale withcanonical decomposition (208) Since the canonical decom-position of a continuous semimartingale is unique up toindistinguishability we deduce that 119861 = 119872 = 0 up toindistinguishability Using the Ito isometry (39) we obtain P-almost surely
int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩
+ ⟨120585 120572 (119883119904) minus 119863120601 (119884
showing that119883 is a local strong solution to (44) with lifetime120591
(3)rArr(1) This implication is a direct consequence ofProposition 29
The results from this section are closely related to the exis-tence of finite dimensional realizations that is the existenceof invariantmanifolds for each starting point ℎ
0 andwe point
out the papers [17ndash22] regarding this topic Furthermore wemention thatTheorem 68 has been extended in [23] to SPDEswith jumps
Acknowledgments
The author is grateful to Daniel Gaigall Georg GrafendorferFlorian Modler and Thomas Salfeld for the valuable com-ments and discussions
References
[1] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions CambridgeUniversity Press CambridgeUK 1992
[2] C Prevot and M Rockner A Concise Course on StochasticPartial Differential Equations Springer Berlin Germany 2007
[3] L Gawarecki and V Mandrekar Stochastic Differential Equa-tions in Infinite Dimensions with Applications to StochasticPartial Differential Equations Springer Berlin Germany 2011
[4] S Peszat and J Zabczyk Stochastic Partial Differential Equationswith Levy Noise Cambridge University Press Cambridge UK2007
[5] S Albeverio V Mandrekar and B Rudiger ldquoExistence of mildsolutions for stochastic differential equations and semilinearequations with non-Gaussian Levy noiserdquo Stochastic Processesand Their Applications vol 119 no 3 pp 835ndash863 2009
[6] D Filipovic S Tappe and J Teichmann ldquoJump-diffusions inHilbert spaces existence stability and numericsrdquo Stochasticsvol 82 no 5 pp 475ndash520 2010
[7] C Marinelli C Prevot and M Rockner ldquoRegular dependenceon initial data for stochastic evolution equations with multi-plicative Poisson noiserdquo Journal of Functional Analysis vol 258no 2 pp 616ndash649 2010
[8] S Tappe ldquoSome refinements of existence results for SPDEsdriven by Wiener processes and Poisson random measuresrdquoInternational Journal of Stochastic Analysis vol 2012 Article ID236327 24 pages 2012
[9] T Nakayama ldquoSupport theorem for mild solutions of SDErsquos inHilbert spacesrdquo Journal of Mathematical Sciences the Universityof Tokyo vol 11 no 3 pp 245ndash311 2004
[10] T Nakayama ldquoViability theorem for SPDErsquos including HJMframeworkrdquo Journal of Mathematical Sciences the University ofTokyo vol 11 no 3 pp 313ndash324 2004
[11] W Rudin Functional Analysis McGraw-Hill New York NYUSA 2nd edition 1991
[12] DWerner Funktionalanalysis Springer Berlin Germany 2007[13] C D Aliprantis and K C Border Infinite Dimensional Analysis
Springer Berlin Germany 3rd edition 2006[14] S N Ethier and T G KurtzMarkov Processes Characterization
andConvergenceWiley Series in Probability and Statistics JohnWiley amp Sons New Jersey NJ USA 1986
[15] V Mandrekar and B Rudiger ldquoExistence and uniqueness ofpath wise solutions for stochastic integral equations driven byLevy noise on separable Banach spacesrdquo Stochastics vol 78 no4 pp 189ndash212 2006
[16] D Filipovic Consistency Problems for Heath-Jarrow-MortonInterest Rate Models Springer Berlin Germany 2001
[17] T Bjork and L Svensson ldquoOn the existence of finite-dimensional realizations for nonlinear forward rate modelsrdquoMathematical Finance vol 11 no 2 pp 205ndash243 2001
[18] T Bjork and C Landen ldquoOn the construction of finite dimen-sional realizations for nonlinear forward rate modelsrdquo Financeand Stochastics vol 6 no 3 pp 303ndash331 2002
[19] D Filipovic and J Teichmann ldquoExistence of invariantmanifoldsfor stochastic equations in infinite dimensionrdquo Journal ofFunctional Analysis vol 197 no 2 pp 398ndash432 2003
[20] D Filipovic and J Teichmann ldquoOn the geometry of the termstructure of interest ratesrdquo Proceedings of The Royal Society ofLondon A vol 460 no 2041 pp 129ndash167 2004
[21] S Tappe ldquoAn alternative approach on the existence of affinerealizations for HJM term structure modelsrdquo Proceedings of
International Journal of Stochastic Analysis 25
TheRoyal Society of LondonA vol 466 no 2122 pp 3033ndash30602010
[22] S Tappe ldquoExistence of affine realizations for Levy term struc-ture modelsrdquo Proceedings of The Royal Society of London A vol468 no 2147 pp 3685ndash3704 2012
[23] D Filipovic S Tappe and J Teichmann ldquoInvariant manifoldswith boundary for jump-diffusionsrdquo httparxivorgabs12021076
Consequently there exists an index 119899 isin N such that Φ119899 is
a contraction and hence according to the extension of theBanach fixed point theorem (see Corollary 46) the mappingΦ has a unique fixed point 119883 isin 119871
2119901
119879(119867) This fixed point 119883 is
a solution to the variation of constants equation (130) Since119879 ge 0 was arbitrary there exists a process 119883 which is a solu-tion of the variation of constants equation
119883119905= 119878
119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 + int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904
119905 ge 0
(146)
Step 14 In order to prove that 119883 is a mild solution to (44)it remains to show that 119883 has a continuous version ByLemma 12 the process
119905 997891997888rarr 119878119905ℎ0 119905 ge 0 (147)
is continuous and by Proposition 41 the process
int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904 119905 ge 0 (148)
is continuous too Moreover for every 119879 ge 0 we have by thelinear growth condition (128) Holderrsquos inequality and since119883 isin 119871
Thus by Proposition 43 the stochastic convolution 120590 ⋆ 119882
given by
(120590 ⋆ 119882)119905= int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0 (150)
has a continuous version and consequently the process119883hasa continuous version too This continuous version is a mildsolution to (44)
Step 2 Now let ℎ0 Ω rarr 119867 be an arbitraryF
0-measurable
random variable We define the sequence (ℎ119899)119899isinN of F
0-
measurable random variables as
ℎ119899
0= ℎ
01ℎ0le119899
119899 isin N (151)
Let 119899 isin N be arbitrary Then as ℎ119899
0is bounded we have
E[ℎ119899
02119901
] lt infin for all 119901 gt 1 By Step 1 the SPDE
119889119883119899
119905= (119860119883
119899
119905+ 120572 (119905 119883
119899
119905)) 119889119905 + 120590 (119905 119883
119899
119905) 119889119882
119905
119883119899
0= ℎ
119899
0
(152)
18 International Journal of Stochastic Analysis
has amild solution119883119899We define the sequence (Ω
119899)119899isinN sub F
0
as
Ω119899=
1003817100381710038171003817ℎ01003817100381710038171003817 le 119899 119899 isin N (153)
Then we have Ω119899sub Ω
119898for 119899 le 119898 Ω = ⋃
119899isinN Ω119899 and
Ω119899sub ℎ
119899
0= ℎ
119898
0 sub ℎ
119899
0= ℎ
0 forall119899 le 119898 (154)
Thus byTheorem 48 we have (up to indistinguishability)
119883119899
1Ω119899
= 119883119898
1Ω119899
forall119899 le 119898 (155)
Consequently the process
119883 = lim119899rarrinfin
119883119899
1Ω119899
(156)
is a well-defined continuous and adapted process and wehave
119883119899
1Ω119899
= 119883119898
1Ω119899
= 1198831Ω119899
forall119899 le 119898 (157)
Furthermore we obtain P-almost surely
119883119905= lim
119899rarrinfin
119883119899
1199051Ω119899
= lim119899rarrinfin
1Ω119899
(119878119905ℎ119899
0+ int
119905
0
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+int
119905
0
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ119899
0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= lim119899rarrinfin
1Ω119899
(119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= 119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(158)
proving that 119883 is a mild solution to (44)
Remark 51 For the proof of Theorem 50 we have usedCorollary 46 which is a slight extension of the Banach fixedpoint theorem Such an idea has been applied for example in[15]
Remark 52 A recent method for proving existence anduniqueness of mild solutions to the SPDE (44) is the methodof the moving frame presented in [6] see also [8] It allowsto reduce SPDE problems to the study of SDEs in infinitedimension In order to apply this method we need that thesemigroup (119878
119905)119905ge0
is a semigroup of pseudocontractions
We close this section with a consequence about theexistence of weak solutions
Corollary 53 Suppose that conditions (125)ndash(128) are ful-filled Let ℎ
0 Ω rarr 119867 be aF
0-measurable random variable
such that E[ℎ02119901
] lt infin for some 119901 gt 1 Then there exists a(up to indistinguishability) unique weak solution 119883 to (44)
Proof According to Proposition 35 every weak solution119883 to(44) is also a mild solution to (44)Therefore the uniquenessof weak solutions to (44) is a consequence of Theorem 48
It remains to prove the existence of a weak solution to(44) Let 119879 ge 0 be arbitrary By Theorem 50 and its proofthere exists a mild solution119883 isin 119871
2119901
119879(119867) to (44) By the linear
growth condition (128) and Holderrsquos inequality we obtain
showing that condition (72) is fulfilled Thus byProposition 37 the process 119883 is also a weak solution to(44)
8 Invariant Manifolds for WeakSolutions to SPDEs
In this section we deal with invariant manifolds for time-homogeneous SPDEs of the type (44) This topic arises fromthe natural desire to express the solutions of the SPDE (44)which generally live in the infinite dimensional Hilbert space119867 by means of a finite dimensional state process and thusto ensure larger analytical tractability Our goal is to findconditions on the generator 119860 and the coefficients 120572 120590 suchthat for every starting point of a finite dimensional submani-fold the solution process stays on the submanifold
We start with the required preliminaries about finitedimensional submanifolds inHilbert spaces In the sequel let119867 be a separable Hilbert space
Definition 54 Let 119898 119896 isin N be positive integers A subsetM sub 119867 is called an 119898-dimensional 119862119896-submanifold of 119867
International Journal of Stochastic Analysis 19
if for every ℎ isin M there exist an open neighborhood 119880 sub 119867
of ℎ an open set 119881 sub R119898 and a mapping 120601 isin 1198622
(119881119867) suchthat
(1) the mapping 120601 119881 rarr 119880 cap M is a homeomorphism(2) for all 119910 isin 119881 the mapping 119863120601(119910) is injective
The mapping 120601 is called a parametrization ofM around ℎ
In what follows let M be an 119898-dimensional 119862119896-
submanifold of 119867
Lemma 55 Let 120601119894
119881119894
rarr 119880119894cap M 119894 = 1 2 be two
parametrizations with119882= 1198801cap119880
2capM = 0Then themapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (160)
is a 119862119896-diffeomorphism
Proof See [16 Lemma 611]
Corollary 56 Let ℎ isin M be arbitrary and let 120601119894
119881119894
rarr
119880119894capM 119894 = 1 2 be two parametrizations ofM around ℎ Then
one has
1198631206011(119910
1) (R
119898
) = 1198631206012(119910
2) (R
119898
) (161)
where 119910119894= 120601
minus1
119894(ℎ) for 119894 = 1 2
Proof Since1198801and119880
2are open neighborhoods of ℎ we have
119882= 1198801cap 119880
2cap M = 0 Thus by Lemma 55 the mapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (162)
is a 119862119896-diffeomorphism Using the chain rule we obtain
1198631206012(119910
2) (R
119898
)
= 119863 (1206011∘ (120601
minus1
1∘ 120601
2)) (119910
2) (R
119898
)
= 1198631206011(119910
1)119863 (120601
minus1
1∘ 120601
2) (119910
2) (R
119898
)
sub 1198631206011(119910
1) (R
119898
)
(163)
and analogously we prove that 1198631206011(119910
1)(R119898
) sub
1198631206012(119910
2)(R119898
)
Definition 57 Let ℎ isin M be arbitraryThe tangent space ofMto ℎ is the subspace
119879ℎM = 119863120601 (119910) (R
119898
) (164)
where 119910 = 120601minus1
(ℎ) and 120601 119881 rarr 119880 cap M denotes aparametrization ofM around ℎ
Remark 58 Note that according to Corollary 56 theDefinition 57 of the tangent space 119879
ℎM does not depend on
the choice of the parametrization 120601 119881 rarr 119880 cap M
Proposition 59 Let ℎ isin M be arbitrary and let 120601 119881 rarr
119880 cap M be a parametrization ofM around ℎ Then there existan open set 119881
0sub 119881 an open neighborhood 119880
0sub 119880 of ℎ and
a mapping 120601 isin 119862119896
119887(R119898
119867) with 120601|1198810
= 120601|1198810
such that 120601|1198810
1198810
rarr 1198800cap M is a parametrization ofM around ℎ too
Proof See [16 Remark 611]
Remark 60 By Proposition 59 we may assume that anyparametrization 120601 119881 rarr 119880 cap M has an extension 120601 isin
119862119896
119887(R119898
119867)
Proposition 61 Let 119863 sub 119867 be a dense subset For every ℎ0isin
M there exist 1205771 120577
119898isin 119863 and a parametrization 120601 119881 rarr
119880 cap M around ℎ0such that
120601 (⟨120577 ℎ⟩) = ℎ forallℎ isin 119880 cap M (165)
where one uses the notation ⟨120577 ℎ⟩ = (⟨1205771 ℎ⟩ ⟨120577
119898 ℎ⟩) isin
R119898
Proof See [16 Proposition 612]
Proposition 62 Let 120601 119881 rarr 119880 cap M be a parametrizationas in Proposition 61 Then the following statements are true
(1) The elements 1205771 120577
119898are linearly independent in 119867
(2) For every ℎ isin 119880 cap M one has the direct sum decom-position
Therefore by Proposition 64 for every 1199100isin R119898 there exists
a (up to indistinguishability) unique weak solution to (192)which according to Proposition 39 is also a strong solutionto (192) The uniqueness of strong solutions to (192) is a con-sequence of Proposition 39 andTheorem 48
Now we are ready to formulate and prove ourmain resultof this section
Theorem 68 The following statements are equivalent
(1) The submanifoldM is locally invariant for (176)(2) One has
M sub D (119860) (195)
120590119895
(ℎ) isin 119879ℎM forallℎ isin M 119886119897119897 119895 isin N (196)
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) isin 119879ℎM forallℎ isin M (197)
(3) The operator 119860 is continuous onM and for each ℎ0isin
M there exists a local strong solution 119883 to (176) withsome lifetime 120591 gt 0 such that
119883119905and120591
isin M forall119905 ge 0 P-119886119897119898119900119904119905 119904119906119903119890119897119910 (198)
Proof (1)rArr(2) Let ℎ isin M be arbitrary By Proposition 61and Remark 60 there exist elements 120577
1 120577
119898isin D(119860
lowast
) anda parametrization 120601 119881 rarr 119880 cap M around ℎ such that theinverse 120601
minus1
119880 cap M rarr 119881 is given by 120601minus1
= ⟨120577 ∙⟩ and 120601
has an extension 120601 isin 1198622
119887(R119898
119867) Since the submanifoldM islocally invariant for (176) there exists a local weak solution119883
to (176) with initial condition ℎ and some lifetime 984858 gt 0 suchthat
119883119905and984858
isin M forall119905 ge 0 P-almost surely (199)
Since119880 is an open neighborhood of ℎ there exists 120598 gt 0 suchthat 119861
120598(ℎ) sub 119880 where 119861
120598(ℎ) denotes the open ball
119861120598(ℎ) = 119892 isin 119867
1003817100381710038171003817119892 minus ℎ1003817100381710038171003817 lt 120598 (200)
We define the stopping time
120591 = 984858 and inf 119905 ge 0 119883119905notin 119861
120598(ℎ) (201)
Since the process119883 has continuous sample paths and satisfies1198830= ℎ we have 120591 gt 0 and P-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (202)
22 International Journal of Stochastic Analysis
Defining the R119898-valued process 119884 = ⟨120577 119883⟩ we have P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (203)
Moreover since 119883 is a weak solution to (176) with initialcondition ℎ setting 119910 = ⟨120577 ℎ⟩ isin 119881 we have P-almost surely
119884119905and120591
= ⟨120577 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120577 120590119895
(119883119904)⟩ 119889120573
119895
119904
= ⟨120577 ℎ⟩ + int
119905and120591
0
120572120601120577
(⟨120577 119883119904⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(⟨120577 119883
119904⟩) 119889120573
119895
119904
= 119910 + int
119905and120591
0
120572120601120577
(119884119904) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(204)
showing that 119884 is a local strong solution to (192) with initialcondition 119910 By Itorsquos formula (Theorem 26) we obtain P-almost surely
119883119905and120591
= 120601 (119884119905and120591
)
= ℎ + int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904)))119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(205)
Now let 120585 isin D(119860lowast
) be arbitrary Then we have P-almostsurely
⟨120585 119883119905and120591
⟩
= ⟨120585 ℎ⟩
+ int
119905and120591
0
⟨120585119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 119863120601 (119884119904) 120590
119895
120601120577(119884
119904)⟩119889120573
119895
119904 119905 ge 0
(206)
On the other hand since 119883 is a local weak solution to (176)with initial condition ℎ and lifetime 120591 we have P-almostsurely for all 119905 ge 0 the identity
⟨120585 119883119905and120591
⟩ = ⟨120585 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904)⟩ 119889120573
119895
119904
(207)
Combining (206) and (207) yields up to indistinguishability
119861 + 119872 = 0 (208)
where the processes 119861 and 119872 are defined as
119861119905= int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩) 119889119904
119905 ge 0
119872119905= sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩ 119889120573
119895
119904
119905 ge 0
(209)
The process 119861 + 119872 is a continuous semimartingale withcanonical decomposition (208) Since the canonical decom-position of a continuous semimartingale is unique up toindistinguishability we deduce that 119861 = 119872 = 0 up toindistinguishability Using the Ito isometry (39) we obtain P-almost surely
int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩
+ ⟨120585 120572 (119883119904) minus 119863120601 (119884
showing that119883 is a local strong solution to (44) with lifetime120591
(3)rArr(1) This implication is a direct consequence ofProposition 29
The results from this section are closely related to the exis-tence of finite dimensional realizations that is the existenceof invariantmanifolds for each starting point ℎ
0 andwe point
out the papers [17ndash22] regarding this topic Furthermore wemention thatTheorem 68 has been extended in [23] to SPDEswith jumps
Acknowledgments
The author is grateful to Daniel Gaigall Georg GrafendorferFlorian Modler and Thomas Salfeld for the valuable com-ments and discussions
References
[1] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions CambridgeUniversity Press CambridgeUK 1992
[2] C Prevot and M Rockner A Concise Course on StochasticPartial Differential Equations Springer Berlin Germany 2007
[3] L Gawarecki and V Mandrekar Stochastic Differential Equa-tions in Infinite Dimensions with Applications to StochasticPartial Differential Equations Springer Berlin Germany 2011
[4] S Peszat and J Zabczyk Stochastic Partial Differential Equationswith Levy Noise Cambridge University Press Cambridge UK2007
[5] S Albeverio V Mandrekar and B Rudiger ldquoExistence of mildsolutions for stochastic differential equations and semilinearequations with non-Gaussian Levy noiserdquo Stochastic Processesand Their Applications vol 119 no 3 pp 835ndash863 2009
[6] D Filipovic S Tappe and J Teichmann ldquoJump-diffusions inHilbert spaces existence stability and numericsrdquo Stochasticsvol 82 no 5 pp 475ndash520 2010
[7] C Marinelli C Prevot and M Rockner ldquoRegular dependenceon initial data for stochastic evolution equations with multi-plicative Poisson noiserdquo Journal of Functional Analysis vol 258no 2 pp 616ndash649 2010
[8] S Tappe ldquoSome refinements of existence results for SPDEsdriven by Wiener processes and Poisson random measuresrdquoInternational Journal of Stochastic Analysis vol 2012 Article ID236327 24 pages 2012
[9] T Nakayama ldquoSupport theorem for mild solutions of SDErsquos inHilbert spacesrdquo Journal of Mathematical Sciences the Universityof Tokyo vol 11 no 3 pp 245ndash311 2004
[10] T Nakayama ldquoViability theorem for SPDErsquos including HJMframeworkrdquo Journal of Mathematical Sciences the University ofTokyo vol 11 no 3 pp 313ndash324 2004
[11] W Rudin Functional Analysis McGraw-Hill New York NYUSA 2nd edition 1991
[12] DWerner Funktionalanalysis Springer Berlin Germany 2007[13] C D Aliprantis and K C Border Infinite Dimensional Analysis
Springer Berlin Germany 3rd edition 2006[14] S N Ethier and T G KurtzMarkov Processes Characterization
andConvergenceWiley Series in Probability and Statistics JohnWiley amp Sons New Jersey NJ USA 1986
[15] V Mandrekar and B Rudiger ldquoExistence and uniqueness ofpath wise solutions for stochastic integral equations driven byLevy noise on separable Banach spacesrdquo Stochastics vol 78 no4 pp 189ndash212 2006
[16] D Filipovic Consistency Problems for Heath-Jarrow-MortonInterest Rate Models Springer Berlin Germany 2001
[17] T Bjork and L Svensson ldquoOn the existence of finite-dimensional realizations for nonlinear forward rate modelsrdquoMathematical Finance vol 11 no 2 pp 205ndash243 2001
[18] T Bjork and C Landen ldquoOn the construction of finite dimen-sional realizations for nonlinear forward rate modelsrdquo Financeand Stochastics vol 6 no 3 pp 303ndash331 2002
[19] D Filipovic and J Teichmann ldquoExistence of invariantmanifoldsfor stochastic equations in infinite dimensionrdquo Journal ofFunctional Analysis vol 197 no 2 pp 398ndash432 2003
[20] D Filipovic and J Teichmann ldquoOn the geometry of the termstructure of interest ratesrdquo Proceedings of The Royal Society ofLondon A vol 460 no 2041 pp 129ndash167 2004
[21] S Tappe ldquoAn alternative approach on the existence of affinerealizations for HJM term structure modelsrdquo Proceedings of
International Journal of Stochastic Analysis 25
TheRoyal Society of LondonA vol 466 no 2122 pp 3033ndash30602010
[22] S Tappe ldquoExistence of affine realizations for Levy term struc-ture modelsrdquo Proceedings of The Royal Society of London A vol468 no 2147 pp 3685ndash3704 2012
[23] D Filipovic S Tappe and J Teichmann ldquoInvariant manifoldswith boundary for jump-diffusionsrdquo httparxivorgabs12021076
has amild solution119883119899We define the sequence (Ω
119899)119899isinN sub F
0
as
Ω119899=
1003817100381710038171003817ℎ01003817100381710038171003817 le 119899 119899 isin N (153)
Then we have Ω119899sub Ω
119898for 119899 le 119898 Ω = ⋃
119899isinN Ω119899 and
Ω119899sub ℎ
119899
0= ℎ
119898
0 sub ℎ
119899
0= ℎ
0 forall119899 le 119898 (154)
Thus byTheorem 48 we have (up to indistinguishability)
119883119899
1Ω119899
= 119883119898
1Ω119899
forall119899 le 119898 (155)
Consequently the process
119883 = lim119899rarrinfin
119883119899
1Ω119899
(156)
is a well-defined continuous and adapted process and wehave
119883119899
1Ω119899
= 119883119898
1Ω119899
= 1198831Ω119899
forall119899 le 119898 (157)
Furthermore we obtain P-almost surely
119883119905= lim
119899rarrinfin
119883119899
1199051Ω119899
= lim119899rarrinfin
1Ω119899
(119878119905ℎ119899
0+ int
119905
0
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+int
119905
0
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ119899
0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119899
119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119899
119904) 119889119882
119904)
= lim119899rarrinfin
(119878119905(1
Ω119899
ℎ0) + int
119905
0
1Ω119899
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
1Ω119899
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= lim119899rarrinfin
1Ω119899
(119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904)
= 119878119905ℎ0+ int
119905
0
119878119905minus119904
120572 (119904 119883119904) 119889119904
+ int
119905
0
119878119905minus119904
120590 (119904 119883119904) 119889119882
119904 119905 ge 0
(158)
proving that 119883 is a mild solution to (44)
Remark 51 For the proof of Theorem 50 we have usedCorollary 46 which is a slight extension of the Banach fixedpoint theorem Such an idea has been applied for example in[15]
Remark 52 A recent method for proving existence anduniqueness of mild solutions to the SPDE (44) is the methodof the moving frame presented in [6] see also [8] It allowsto reduce SPDE problems to the study of SDEs in infinitedimension In order to apply this method we need that thesemigroup (119878
119905)119905ge0
is a semigroup of pseudocontractions
We close this section with a consequence about theexistence of weak solutions
Corollary 53 Suppose that conditions (125)ndash(128) are ful-filled Let ℎ
0 Ω rarr 119867 be aF
0-measurable random variable
such that E[ℎ02119901
] lt infin for some 119901 gt 1 Then there exists a(up to indistinguishability) unique weak solution 119883 to (44)
Proof According to Proposition 35 every weak solution119883 to(44) is also a mild solution to (44)Therefore the uniquenessof weak solutions to (44) is a consequence of Theorem 48
It remains to prove the existence of a weak solution to(44) Let 119879 ge 0 be arbitrary By Theorem 50 and its proofthere exists a mild solution119883 isin 119871
2119901
119879(119867) to (44) By the linear
growth condition (128) and Holderrsquos inequality we obtain
showing that condition (72) is fulfilled Thus byProposition 37 the process 119883 is also a weak solution to(44)
8 Invariant Manifolds for WeakSolutions to SPDEs
In this section we deal with invariant manifolds for time-homogeneous SPDEs of the type (44) This topic arises fromthe natural desire to express the solutions of the SPDE (44)which generally live in the infinite dimensional Hilbert space119867 by means of a finite dimensional state process and thusto ensure larger analytical tractability Our goal is to findconditions on the generator 119860 and the coefficients 120572 120590 suchthat for every starting point of a finite dimensional submani-fold the solution process stays on the submanifold
We start with the required preliminaries about finitedimensional submanifolds inHilbert spaces In the sequel let119867 be a separable Hilbert space
Definition 54 Let 119898 119896 isin N be positive integers A subsetM sub 119867 is called an 119898-dimensional 119862119896-submanifold of 119867
International Journal of Stochastic Analysis 19
if for every ℎ isin M there exist an open neighborhood 119880 sub 119867
of ℎ an open set 119881 sub R119898 and a mapping 120601 isin 1198622
(119881119867) suchthat
(1) the mapping 120601 119881 rarr 119880 cap M is a homeomorphism(2) for all 119910 isin 119881 the mapping 119863120601(119910) is injective
The mapping 120601 is called a parametrization ofM around ℎ
In what follows let M be an 119898-dimensional 119862119896-
submanifold of 119867
Lemma 55 Let 120601119894
119881119894
rarr 119880119894cap M 119894 = 1 2 be two
parametrizations with119882= 1198801cap119880
2capM = 0Then themapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (160)
is a 119862119896-diffeomorphism
Proof See [16 Lemma 611]
Corollary 56 Let ℎ isin M be arbitrary and let 120601119894
119881119894
rarr
119880119894capM 119894 = 1 2 be two parametrizations ofM around ℎ Then
one has
1198631206011(119910
1) (R
119898
) = 1198631206012(119910
2) (R
119898
) (161)
where 119910119894= 120601
minus1
119894(ℎ) for 119894 = 1 2
Proof Since1198801and119880
2are open neighborhoods of ℎ we have
119882= 1198801cap 119880
2cap M = 0 Thus by Lemma 55 the mapping
120601minus1
1∘ 120601
2 120601
minus1
2(119882) 997888rarr 120601
minus1
1(119882) (162)
is a 119862119896-diffeomorphism Using the chain rule we obtain
1198631206012(119910
2) (R
119898
)
= 119863 (1206011∘ (120601
minus1
1∘ 120601
2)) (119910
2) (R
119898
)
= 1198631206011(119910
1)119863 (120601
minus1
1∘ 120601
2) (119910
2) (R
119898
)
sub 1198631206011(119910
1) (R
119898
)
(163)
and analogously we prove that 1198631206011(119910
1)(R119898
) sub
1198631206012(119910
2)(R119898
)
Definition 57 Let ℎ isin M be arbitraryThe tangent space ofMto ℎ is the subspace
119879ℎM = 119863120601 (119910) (R
119898
) (164)
where 119910 = 120601minus1
(ℎ) and 120601 119881 rarr 119880 cap M denotes aparametrization ofM around ℎ
Remark 58 Note that according to Corollary 56 theDefinition 57 of the tangent space 119879
ℎM does not depend on
the choice of the parametrization 120601 119881 rarr 119880 cap M
Proposition 59 Let ℎ isin M be arbitrary and let 120601 119881 rarr
119880 cap M be a parametrization ofM around ℎ Then there existan open set 119881
0sub 119881 an open neighborhood 119880
0sub 119880 of ℎ and
a mapping 120601 isin 119862119896
119887(R119898
119867) with 120601|1198810
= 120601|1198810
such that 120601|1198810
1198810
rarr 1198800cap M is a parametrization ofM around ℎ too
Proof See [16 Remark 611]
Remark 60 By Proposition 59 we may assume that anyparametrization 120601 119881 rarr 119880 cap M has an extension 120601 isin
119862119896
119887(R119898
119867)
Proposition 61 Let 119863 sub 119867 be a dense subset For every ℎ0isin
M there exist 1205771 120577
119898isin 119863 and a parametrization 120601 119881 rarr
119880 cap M around ℎ0such that
120601 (⟨120577 ℎ⟩) = ℎ forallℎ isin 119880 cap M (165)
where one uses the notation ⟨120577 ℎ⟩ = (⟨1205771 ℎ⟩ ⟨120577
119898 ℎ⟩) isin
R119898
Proof See [16 Proposition 612]
Proposition 62 Let 120601 119881 rarr 119880 cap M be a parametrizationas in Proposition 61 Then the following statements are true
(1) The elements 1205771 120577
119898are linearly independent in 119867
(2) For every ℎ isin 119880 cap M one has the direct sum decom-position
Therefore by Proposition 64 for every 1199100isin R119898 there exists
a (up to indistinguishability) unique weak solution to (192)which according to Proposition 39 is also a strong solutionto (192) The uniqueness of strong solutions to (192) is a con-sequence of Proposition 39 andTheorem 48
Now we are ready to formulate and prove ourmain resultof this section
Theorem 68 The following statements are equivalent
(1) The submanifoldM is locally invariant for (176)(2) One has
M sub D (119860) (195)
120590119895
(ℎ) isin 119879ℎM forallℎ isin M 119886119897119897 119895 isin N (196)
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) isin 119879ℎM forallℎ isin M (197)
(3) The operator 119860 is continuous onM and for each ℎ0isin
M there exists a local strong solution 119883 to (176) withsome lifetime 120591 gt 0 such that
119883119905and120591
isin M forall119905 ge 0 P-119886119897119898119900119904119905 119904119906119903119890119897119910 (198)
Proof (1)rArr(2) Let ℎ isin M be arbitrary By Proposition 61and Remark 60 there exist elements 120577
1 120577
119898isin D(119860
lowast
) anda parametrization 120601 119881 rarr 119880 cap M around ℎ such that theinverse 120601
minus1
119880 cap M rarr 119881 is given by 120601minus1
= ⟨120577 ∙⟩ and 120601
has an extension 120601 isin 1198622
119887(R119898
119867) Since the submanifoldM islocally invariant for (176) there exists a local weak solution119883
to (176) with initial condition ℎ and some lifetime 984858 gt 0 suchthat
119883119905and984858
isin M forall119905 ge 0 P-almost surely (199)
Since119880 is an open neighborhood of ℎ there exists 120598 gt 0 suchthat 119861
120598(ℎ) sub 119880 where 119861
120598(ℎ) denotes the open ball
119861120598(ℎ) = 119892 isin 119867
1003817100381710038171003817119892 minus ℎ1003817100381710038171003817 lt 120598 (200)
We define the stopping time
120591 = 984858 and inf 119905 ge 0 119883119905notin 119861
120598(ℎ) (201)
Since the process119883 has continuous sample paths and satisfies1198830= ℎ we have 120591 gt 0 and P-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (202)
22 International Journal of Stochastic Analysis
Defining the R119898-valued process 119884 = ⟨120577 119883⟩ we have P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (203)
Moreover since 119883 is a weak solution to (176) with initialcondition ℎ setting 119910 = ⟨120577 ℎ⟩ isin 119881 we have P-almost surely
119884119905and120591
= ⟨120577 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120577 120590119895
(119883119904)⟩ 119889120573
119895
119904
= ⟨120577 ℎ⟩ + int
119905and120591
0
120572120601120577
(⟨120577 119883119904⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(⟨120577 119883
119904⟩) 119889120573
119895
119904
= 119910 + int
119905and120591
0
120572120601120577
(119884119904) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(204)
showing that 119884 is a local strong solution to (192) with initialcondition 119910 By Itorsquos formula (Theorem 26) we obtain P-almost surely
119883119905and120591
= 120601 (119884119905and120591
)
= ℎ + int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904)))119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(205)
Now let 120585 isin D(119860lowast
) be arbitrary Then we have P-almostsurely
⟨120585 119883119905and120591
⟩
= ⟨120585 ℎ⟩
+ int
119905and120591
0
⟨120585119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 119863120601 (119884119904) 120590
119895
120601120577(119884
119904)⟩119889120573
119895
119904 119905 ge 0
(206)
On the other hand since 119883 is a local weak solution to (176)with initial condition ℎ and lifetime 120591 we have P-almostsurely for all 119905 ge 0 the identity
⟨120585 119883119905and120591
⟩ = ⟨120585 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904)⟩ 119889120573
119895
119904
(207)
Combining (206) and (207) yields up to indistinguishability
119861 + 119872 = 0 (208)
where the processes 119861 and 119872 are defined as
119861119905= int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩) 119889119904
119905 ge 0
119872119905= sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩ 119889120573
119895
119904
119905 ge 0
(209)
The process 119861 + 119872 is a continuous semimartingale withcanonical decomposition (208) Since the canonical decom-position of a continuous semimartingale is unique up toindistinguishability we deduce that 119861 = 119872 = 0 up toindistinguishability Using the Ito isometry (39) we obtain P-almost surely
int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩
+ ⟨120585 120572 (119883119904) minus 119863120601 (119884
showing that119883 is a local strong solution to (44) with lifetime120591
(3)rArr(1) This implication is a direct consequence ofProposition 29
The results from this section are closely related to the exis-tence of finite dimensional realizations that is the existenceof invariantmanifolds for each starting point ℎ
0 andwe point
out the papers [17ndash22] regarding this topic Furthermore wemention thatTheorem 68 has been extended in [23] to SPDEswith jumps
Acknowledgments
The author is grateful to Daniel Gaigall Georg GrafendorferFlorian Modler and Thomas Salfeld for the valuable com-ments and discussions
References
[1] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions CambridgeUniversity Press CambridgeUK 1992
[2] C Prevot and M Rockner A Concise Course on StochasticPartial Differential Equations Springer Berlin Germany 2007
[3] L Gawarecki and V Mandrekar Stochastic Differential Equa-tions in Infinite Dimensions with Applications to StochasticPartial Differential Equations Springer Berlin Germany 2011
[4] S Peszat and J Zabczyk Stochastic Partial Differential Equationswith Levy Noise Cambridge University Press Cambridge UK2007
[5] S Albeverio V Mandrekar and B Rudiger ldquoExistence of mildsolutions for stochastic differential equations and semilinearequations with non-Gaussian Levy noiserdquo Stochastic Processesand Their Applications vol 119 no 3 pp 835ndash863 2009
[6] D Filipovic S Tappe and J Teichmann ldquoJump-diffusions inHilbert spaces existence stability and numericsrdquo Stochasticsvol 82 no 5 pp 475ndash520 2010
[7] C Marinelli C Prevot and M Rockner ldquoRegular dependenceon initial data for stochastic evolution equations with multi-plicative Poisson noiserdquo Journal of Functional Analysis vol 258no 2 pp 616ndash649 2010
[8] S Tappe ldquoSome refinements of existence results for SPDEsdriven by Wiener processes and Poisson random measuresrdquoInternational Journal of Stochastic Analysis vol 2012 Article ID236327 24 pages 2012
[9] T Nakayama ldquoSupport theorem for mild solutions of SDErsquos inHilbert spacesrdquo Journal of Mathematical Sciences the Universityof Tokyo vol 11 no 3 pp 245ndash311 2004
[10] T Nakayama ldquoViability theorem for SPDErsquos including HJMframeworkrdquo Journal of Mathematical Sciences the University ofTokyo vol 11 no 3 pp 313ndash324 2004
[11] W Rudin Functional Analysis McGraw-Hill New York NYUSA 2nd edition 1991
[12] DWerner Funktionalanalysis Springer Berlin Germany 2007[13] C D Aliprantis and K C Border Infinite Dimensional Analysis
Springer Berlin Germany 3rd edition 2006[14] S N Ethier and T G KurtzMarkov Processes Characterization
andConvergenceWiley Series in Probability and Statistics JohnWiley amp Sons New Jersey NJ USA 1986
[15] V Mandrekar and B Rudiger ldquoExistence and uniqueness ofpath wise solutions for stochastic integral equations driven byLevy noise on separable Banach spacesrdquo Stochastics vol 78 no4 pp 189ndash212 2006
[16] D Filipovic Consistency Problems for Heath-Jarrow-MortonInterest Rate Models Springer Berlin Germany 2001
[17] T Bjork and L Svensson ldquoOn the existence of finite-dimensional realizations for nonlinear forward rate modelsrdquoMathematical Finance vol 11 no 2 pp 205ndash243 2001
[18] T Bjork and C Landen ldquoOn the construction of finite dimen-sional realizations for nonlinear forward rate modelsrdquo Financeand Stochastics vol 6 no 3 pp 303ndash331 2002
[19] D Filipovic and J Teichmann ldquoExistence of invariantmanifoldsfor stochastic equations in infinite dimensionrdquo Journal ofFunctional Analysis vol 197 no 2 pp 398ndash432 2003
[20] D Filipovic and J Teichmann ldquoOn the geometry of the termstructure of interest ratesrdquo Proceedings of The Royal Society ofLondon A vol 460 no 2041 pp 129ndash167 2004
[21] S Tappe ldquoAn alternative approach on the existence of affinerealizations for HJM term structure modelsrdquo Proceedings of
International Journal of Stochastic Analysis 25
TheRoyal Society of LondonA vol 466 no 2122 pp 3033ndash30602010
[22] S Tappe ldquoExistence of affine realizations for Levy term struc-ture modelsrdquo Proceedings of The Royal Society of London A vol468 no 2147 pp 3685ndash3704 2012
[23] D Filipovic S Tappe and J Teichmann ldquoInvariant manifoldswith boundary for jump-diffusionsrdquo httparxivorgabs12021076
Therefore by Proposition 64 for every 1199100isin R119898 there exists
a (up to indistinguishability) unique weak solution to (192)which according to Proposition 39 is also a strong solutionto (192) The uniqueness of strong solutions to (192) is a con-sequence of Proposition 39 andTheorem 48
Now we are ready to formulate and prove ourmain resultof this section
Theorem 68 The following statements are equivalent
(1) The submanifoldM is locally invariant for (176)(2) One has
M sub D (119860) (195)
120590119895
(ℎ) isin 119879ℎM forallℎ isin M 119886119897119897 119895 isin N (196)
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) isin 119879ℎM forallℎ isin M (197)
(3) The operator 119860 is continuous onM and for each ℎ0isin
M there exists a local strong solution 119883 to (176) withsome lifetime 120591 gt 0 such that
119883119905and120591
isin M forall119905 ge 0 P-119886119897119898119900119904119905 119904119906119903119890119897119910 (198)
Proof (1)rArr(2) Let ℎ isin M be arbitrary By Proposition 61and Remark 60 there exist elements 120577
1 120577
119898isin D(119860
lowast
) anda parametrization 120601 119881 rarr 119880 cap M around ℎ such that theinverse 120601
minus1
119880 cap M rarr 119881 is given by 120601minus1
= ⟨120577 ∙⟩ and 120601
has an extension 120601 isin 1198622
119887(R119898
119867) Since the submanifoldM islocally invariant for (176) there exists a local weak solution119883
to (176) with initial condition ℎ and some lifetime 984858 gt 0 suchthat
119883119905and984858
isin M forall119905 ge 0 P-almost surely (199)
Since119880 is an open neighborhood of ℎ there exists 120598 gt 0 suchthat 119861
120598(ℎ) sub 119880 where 119861
120598(ℎ) denotes the open ball
119861120598(ℎ) = 119892 isin 119867
1003817100381710038171003817119892 minus ℎ1003817100381710038171003817 lt 120598 (200)
We define the stopping time
120591 = 984858 and inf 119905 ge 0 119883119905notin 119861
120598(ℎ) (201)
Since the process119883 has continuous sample paths and satisfies1198830= ℎ we have 120591 gt 0 and P-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (202)
22 International Journal of Stochastic Analysis
Defining the R119898-valued process 119884 = ⟨120577 119883⟩ we have P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (203)
Moreover since 119883 is a weak solution to (176) with initialcondition ℎ setting 119910 = ⟨120577 ℎ⟩ isin 119881 we have P-almost surely
119884119905and120591
= ⟨120577 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120577 120590119895
(119883119904)⟩ 119889120573
119895
119904
= ⟨120577 ℎ⟩ + int
119905and120591
0
120572120601120577
(⟨120577 119883119904⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(⟨120577 119883
119904⟩) 119889120573
119895
119904
= 119910 + int
119905and120591
0
120572120601120577
(119884119904) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(204)
showing that 119884 is a local strong solution to (192) with initialcondition 119910 By Itorsquos formula (Theorem 26) we obtain P-almost surely
119883119905and120591
= 120601 (119884119905and120591
)
= ℎ + int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904)))119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(205)
Now let 120585 isin D(119860lowast
) be arbitrary Then we have P-almostsurely
⟨120585 119883119905and120591
⟩
= ⟨120585 ℎ⟩
+ int
119905and120591
0
⟨120585119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 119863120601 (119884119904) 120590
119895
120601120577(119884
119904)⟩119889120573
119895
119904 119905 ge 0
(206)
On the other hand since 119883 is a local weak solution to (176)with initial condition ℎ and lifetime 120591 we have P-almostsurely for all 119905 ge 0 the identity
⟨120585 119883119905and120591
⟩ = ⟨120585 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904)⟩ 119889120573
119895
119904
(207)
Combining (206) and (207) yields up to indistinguishability
119861 + 119872 = 0 (208)
where the processes 119861 and 119872 are defined as
119861119905= int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩) 119889119904
119905 ge 0
119872119905= sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩ 119889120573
119895
119904
119905 ge 0
(209)
The process 119861 + 119872 is a continuous semimartingale withcanonical decomposition (208) Since the canonical decom-position of a continuous semimartingale is unique up toindistinguishability we deduce that 119861 = 119872 = 0 up toindistinguishability Using the Ito isometry (39) we obtain P-almost surely
int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩
+ ⟨120585 120572 (119883119904) minus 119863120601 (119884
showing that119883 is a local strong solution to (44) with lifetime120591
(3)rArr(1) This implication is a direct consequence ofProposition 29
The results from this section are closely related to the exis-tence of finite dimensional realizations that is the existenceof invariantmanifolds for each starting point ℎ
0 andwe point
out the papers [17ndash22] regarding this topic Furthermore wemention thatTheorem 68 has been extended in [23] to SPDEswith jumps
Acknowledgments
The author is grateful to Daniel Gaigall Georg GrafendorferFlorian Modler and Thomas Salfeld for the valuable com-ments and discussions
References
[1] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions CambridgeUniversity Press CambridgeUK 1992
[2] C Prevot and M Rockner A Concise Course on StochasticPartial Differential Equations Springer Berlin Germany 2007
[3] L Gawarecki and V Mandrekar Stochastic Differential Equa-tions in Infinite Dimensions with Applications to StochasticPartial Differential Equations Springer Berlin Germany 2011
[4] S Peszat and J Zabczyk Stochastic Partial Differential Equationswith Levy Noise Cambridge University Press Cambridge UK2007
[5] S Albeverio V Mandrekar and B Rudiger ldquoExistence of mildsolutions for stochastic differential equations and semilinearequations with non-Gaussian Levy noiserdquo Stochastic Processesand Their Applications vol 119 no 3 pp 835ndash863 2009
[6] D Filipovic S Tappe and J Teichmann ldquoJump-diffusions inHilbert spaces existence stability and numericsrdquo Stochasticsvol 82 no 5 pp 475ndash520 2010
[7] C Marinelli C Prevot and M Rockner ldquoRegular dependenceon initial data for stochastic evolution equations with multi-plicative Poisson noiserdquo Journal of Functional Analysis vol 258no 2 pp 616ndash649 2010
[8] S Tappe ldquoSome refinements of existence results for SPDEsdriven by Wiener processes and Poisson random measuresrdquoInternational Journal of Stochastic Analysis vol 2012 Article ID236327 24 pages 2012
[9] T Nakayama ldquoSupport theorem for mild solutions of SDErsquos inHilbert spacesrdquo Journal of Mathematical Sciences the Universityof Tokyo vol 11 no 3 pp 245ndash311 2004
[10] T Nakayama ldquoViability theorem for SPDErsquos including HJMframeworkrdquo Journal of Mathematical Sciences the University ofTokyo vol 11 no 3 pp 313ndash324 2004
[11] W Rudin Functional Analysis McGraw-Hill New York NYUSA 2nd edition 1991
[12] DWerner Funktionalanalysis Springer Berlin Germany 2007[13] C D Aliprantis and K C Border Infinite Dimensional Analysis
Springer Berlin Germany 3rd edition 2006[14] S N Ethier and T G KurtzMarkov Processes Characterization
andConvergenceWiley Series in Probability and Statistics JohnWiley amp Sons New Jersey NJ USA 1986
[15] V Mandrekar and B Rudiger ldquoExistence and uniqueness ofpath wise solutions for stochastic integral equations driven byLevy noise on separable Banach spacesrdquo Stochastics vol 78 no4 pp 189ndash212 2006
[16] D Filipovic Consistency Problems for Heath-Jarrow-MortonInterest Rate Models Springer Berlin Germany 2001
[17] T Bjork and L Svensson ldquoOn the existence of finite-dimensional realizations for nonlinear forward rate modelsrdquoMathematical Finance vol 11 no 2 pp 205ndash243 2001
[18] T Bjork and C Landen ldquoOn the construction of finite dimen-sional realizations for nonlinear forward rate modelsrdquo Financeand Stochastics vol 6 no 3 pp 303ndash331 2002
[19] D Filipovic and J Teichmann ldquoExistence of invariantmanifoldsfor stochastic equations in infinite dimensionrdquo Journal ofFunctional Analysis vol 197 no 2 pp 398ndash432 2003
[20] D Filipovic and J Teichmann ldquoOn the geometry of the termstructure of interest ratesrdquo Proceedings of The Royal Society ofLondon A vol 460 no 2041 pp 129ndash167 2004
[21] S Tappe ldquoAn alternative approach on the existence of affinerealizations for HJM term structure modelsrdquo Proceedings of
International Journal of Stochastic Analysis 25
TheRoyal Society of LondonA vol 466 no 2122 pp 3033ndash30602010
[22] S Tappe ldquoExistence of affine realizations for Levy term struc-ture modelsrdquo Proceedings of The Royal Society of London A vol468 no 2147 pp 3685ndash3704 2012
[23] D Filipovic S Tappe and J Teichmann ldquoInvariant manifoldswith boundary for jump-diffusionsrdquo httparxivorgabs12021076
Therefore by Proposition 64 for every 1199100isin R119898 there exists
a (up to indistinguishability) unique weak solution to (192)which according to Proposition 39 is also a strong solutionto (192) The uniqueness of strong solutions to (192) is a con-sequence of Proposition 39 andTheorem 48
Now we are ready to formulate and prove ourmain resultof this section
Theorem 68 The following statements are equivalent
(1) The submanifoldM is locally invariant for (176)(2) One has
M sub D (119860) (195)
120590119895
(ℎ) isin 119879ℎM forallℎ isin M 119886119897119897 119895 isin N (196)
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) isin 119879ℎM forallℎ isin M (197)
(3) The operator 119860 is continuous onM and for each ℎ0isin
M there exists a local strong solution 119883 to (176) withsome lifetime 120591 gt 0 such that
119883119905and120591
isin M forall119905 ge 0 P-119886119897119898119900119904119905 119904119906119903119890119897119910 (198)
Proof (1)rArr(2) Let ℎ isin M be arbitrary By Proposition 61and Remark 60 there exist elements 120577
1 120577
119898isin D(119860
lowast
) anda parametrization 120601 119881 rarr 119880 cap M around ℎ such that theinverse 120601
minus1
119880 cap M rarr 119881 is given by 120601minus1
= ⟨120577 ∙⟩ and 120601
has an extension 120601 isin 1198622
119887(R119898
119867) Since the submanifoldM islocally invariant for (176) there exists a local weak solution119883
to (176) with initial condition ℎ and some lifetime 984858 gt 0 suchthat
119883119905and984858
isin M forall119905 ge 0 P-almost surely (199)
Since119880 is an open neighborhood of ℎ there exists 120598 gt 0 suchthat 119861
120598(ℎ) sub 119880 where 119861
120598(ℎ) denotes the open ball
119861120598(ℎ) = 119892 isin 119867
1003817100381710038171003817119892 minus ℎ1003817100381710038171003817 lt 120598 (200)
We define the stopping time
120591 = 984858 and inf 119905 ge 0 119883119905notin 119861
120598(ℎ) (201)
Since the process119883 has continuous sample paths and satisfies1198830= ℎ we have 120591 gt 0 and P-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (202)
22 International Journal of Stochastic Analysis
Defining the R119898-valued process 119884 = ⟨120577 119883⟩ we have P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (203)
Moreover since 119883 is a weak solution to (176) with initialcondition ℎ setting 119910 = ⟨120577 ℎ⟩ isin 119881 we have P-almost surely
119884119905and120591
= ⟨120577 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120577 120590119895
(119883119904)⟩ 119889120573
119895
119904
= ⟨120577 ℎ⟩ + int
119905and120591
0
120572120601120577
(⟨120577 119883119904⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(⟨120577 119883
119904⟩) 119889120573
119895
119904
= 119910 + int
119905and120591
0
120572120601120577
(119884119904) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(204)
showing that 119884 is a local strong solution to (192) with initialcondition 119910 By Itorsquos formula (Theorem 26) we obtain P-almost surely
119883119905and120591
= 120601 (119884119905and120591
)
= ℎ + int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904)))119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(205)
Now let 120585 isin D(119860lowast
) be arbitrary Then we have P-almostsurely
⟨120585 119883119905and120591
⟩
= ⟨120585 ℎ⟩
+ int
119905and120591
0
⟨120585119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 119863120601 (119884119904) 120590
119895
120601120577(119884
119904)⟩119889120573
119895
119904 119905 ge 0
(206)
On the other hand since 119883 is a local weak solution to (176)with initial condition ℎ and lifetime 120591 we have P-almostsurely for all 119905 ge 0 the identity
⟨120585 119883119905and120591
⟩ = ⟨120585 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904)⟩ 119889120573
119895
119904
(207)
Combining (206) and (207) yields up to indistinguishability
119861 + 119872 = 0 (208)
where the processes 119861 and 119872 are defined as
119861119905= int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩) 119889119904
119905 ge 0
119872119905= sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩ 119889120573
119895
119904
119905 ge 0
(209)
The process 119861 + 119872 is a continuous semimartingale withcanonical decomposition (208) Since the canonical decom-position of a continuous semimartingale is unique up toindistinguishability we deduce that 119861 = 119872 = 0 up toindistinguishability Using the Ito isometry (39) we obtain P-almost surely
int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩
+ ⟨120585 120572 (119883119904) minus 119863120601 (119884
showing that119883 is a local strong solution to (44) with lifetime120591
(3)rArr(1) This implication is a direct consequence ofProposition 29
The results from this section are closely related to the exis-tence of finite dimensional realizations that is the existenceof invariantmanifolds for each starting point ℎ
0 andwe point
out the papers [17ndash22] regarding this topic Furthermore wemention thatTheorem 68 has been extended in [23] to SPDEswith jumps
Acknowledgments
The author is grateful to Daniel Gaigall Georg GrafendorferFlorian Modler and Thomas Salfeld for the valuable com-ments and discussions
References
[1] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions CambridgeUniversity Press CambridgeUK 1992
[2] C Prevot and M Rockner A Concise Course on StochasticPartial Differential Equations Springer Berlin Germany 2007
[3] L Gawarecki and V Mandrekar Stochastic Differential Equa-tions in Infinite Dimensions with Applications to StochasticPartial Differential Equations Springer Berlin Germany 2011
[4] S Peszat and J Zabczyk Stochastic Partial Differential Equationswith Levy Noise Cambridge University Press Cambridge UK2007
[5] S Albeverio V Mandrekar and B Rudiger ldquoExistence of mildsolutions for stochastic differential equations and semilinearequations with non-Gaussian Levy noiserdquo Stochastic Processesand Their Applications vol 119 no 3 pp 835ndash863 2009
[6] D Filipovic S Tappe and J Teichmann ldquoJump-diffusions inHilbert spaces existence stability and numericsrdquo Stochasticsvol 82 no 5 pp 475ndash520 2010
[7] C Marinelli C Prevot and M Rockner ldquoRegular dependenceon initial data for stochastic evolution equations with multi-plicative Poisson noiserdquo Journal of Functional Analysis vol 258no 2 pp 616ndash649 2010
[8] S Tappe ldquoSome refinements of existence results for SPDEsdriven by Wiener processes and Poisson random measuresrdquoInternational Journal of Stochastic Analysis vol 2012 Article ID236327 24 pages 2012
[9] T Nakayama ldquoSupport theorem for mild solutions of SDErsquos inHilbert spacesrdquo Journal of Mathematical Sciences the Universityof Tokyo vol 11 no 3 pp 245ndash311 2004
[10] T Nakayama ldquoViability theorem for SPDErsquos including HJMframeworkrdquo Journal of Mathematical Sciences the University ofTokyo vol 11 no 3 pp 313ndash324 2004
[11] W Rudin Functional Analysis McGraw-Hill New York NYUSA 2nd edition 1991
[12] DWerner Funktionalanalysis Springer Berlin Germany 2007[13] C D Aliprantis and K C Border Infinite Dimensional Analysis
Springer Berlin Germany 3rd edition 2006[14] S N Ethier and T G KurtzMarkov Processes Characterization
andConvergenceWiley Series in Probability and Statistics JohnWiley amp Sons New Jersey NJ USA 1986
[15] V Mandrekar and B Rudiger ldquoExistence and uniqueness ofpath wise solutions for stochastic integral equations driven byLevy noise on separable Banach spacesrdquo Stochastics vol 78 no4 pp 189ndash212 2006
[16] D Filipovic Consistency Problems for Heath-Jarrow-MortonInterest Rate Models Springer Berlin Germany 2001
[17] T Bjork and L Svensson ldquoOn the existence of finite-dimensional realizations for nonlinear forward rate modelsrdquoMathematical Finance vol 11 no 2 pp 205ndash243 2001
[18] T Bjork and C Landen ldquoOn the construction of finite dimen-sional realizations for nonlinear forward rate modelsrdquo Financeand Stochastics vol 6 no 3 pp 303ndash331 2002
[19] D Filipovic and J Teichmann ldquoExistence of invariantmanifoldsfor stochastic equations in infinite dimensionrdquo Journal ofFunctional Analysis vol 197 no 2 pp 398ndash432 2003
[20] D Filipovic and J Teichmann ldquoOn the geometry of the termstructure of interest ratesrdquo Proceedings of The Royal Society ofLondon A vol 460 no 2041 pp 129ndash167 2004
[21] S Tappe ldquoAn alternative approach on the existence of affinerealizations for HJM term structure modelsrdquo Proceedings of
International Journal of Stochastic Analysis 25
TheRoyal Society of LondonA vol 466 no 2122 pp 3033ndash30602010
[22] S Tappe ldquoExistence of affine realizations for Levy term struc-ture modelsrdquo Proceedings of The Royal Society of London A vol468 no 2147 pp 3685ndash3704 2012
[23] D Filipovic S Tappe and J Teichmann ldquoInvariant manifoldswith boundary for jump-diffusionsrdquo httparxivorgabs12021076
Therefore by Proposition 64 for every 1199100isin R119898 there exists
a (up to indistinguishability) unique weak solution to (192)which according to Proposition 39 is also a strong solutionto (192) The uniqueness of strong solutions to (192) is a con-sequence of Proposition 39 andTheorem 48
Now we are ready to formulate and prove ourmain resultof this section
Theorem 68 The following statements are equivalent
(1) The submanifoldM is locally invariant for (176)(2) One has
M sub D (119860) (195)
120590119895
(ℎ) isin 119879ℎM forallℎ isin M 119886119897119897 119895 isin N (196)
119860ℎ + 120572 (ℎ) minus1
2sum
119895isinN
119863120590119895
(ℎ) 120590119895
(ℎ) isin 119879ℎM forallℎ isin M (197)
(3) The operator 119860 is continuous onM and for each ℎ0isin
M there exists a local strong solution 119883 to (176) withsome lifetime 120591 gt 0 such that
119883119905and120591
isin M forall119905 ge 0 P-119886119897119898119900119904119905 119904119906119903119890119897119910 (198)
Proof (1)rArr(2) Let ℎ isin M be arbitrary By Proposition 61and Remark 60 there exist elements 120577
1 120577
119898isin D(119860
lowast
) anda parametrization 120601 119881 rarr 119880 cap M around ℎ such that theinverse 120601
minus1
119880 cap M rarr 119881 is given by 120601minus1
= ⟨120577 ∙⟩ and 120601
has an extension 120601 isin 1198622
119887(R119898
119867) Since the submanifoldM islocally invariant for (176) there exists a local weak solution119883
to (176) with initial condition ℎ and some lifetime 984858 gt 0 suchthat
119883119905and984858
isin M forall119905 ge 0 P-almost surely (199)
Since119880 is an open neighborhood of ℎ there exists 120598 gt 0 suchthat 119861
120598(ℎ) sub 119880 where 119861
120598(ℎ) denotes the open ball
119861120598(ℎ) = 119892 isin 119867
1003817100381710038171003817119892 minus ℎ1003817100381710038171003817 lt 120598 (200)
We define the stopping time
120591 = 984858 and inf 119905 ge 0 119883119905notin 119861
120598(ℎ) (201)
Since the process119883 has continuous sample paths and satisfies1198830= ℎ we have 120591 gt 0 and P-almost surely
119883119905and120591
isin 119880 cap M forall119905 ge 0 (202)
22 International Journal of Stochastic Analysis
Defining the R119898-valued process 119884 = ⟨120577 119883⟩ we have P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (203)
Moreover since 119883 is a weak solution to (176) with initialcondition ℎ setting 119910 = ⟨120577 ℎ⟩ isin 119881 we have P-almost surely
119884119905and120591
= ⟨120577 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120577 120590119895
(119883119904)⟩ 119889120573
119895
119904
= ⟨120577 ℎ⟩ + int
119905and120591
0
120572120601120577
(⟨120577 119883119904⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(⟨120577 119883
119904⟩) 119889120573
119895
119904
= 119910 + int
119905and120591
0
120572120601120577
(119884119904) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(204)
showing that 119884 is a local strong solution to (192) with initialcondition 119910 By Itorsquos formula (Theorem 26) we obtain P-almost surely
119883119905and120591
= 120601 (119884119905and120591
)
= ℎ + int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904)))119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(205)
Now let 120585 isin D(119860lowast
) be arbitrary Then we have P-almostsurely
⟨120585 119883119905and120591
⟩
= ⟨120585 ℎ⟩
+ int
119905and120591
0
⟨120585119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 119863120601 (119884119904) 120590
119895
120601120577(119884
119904)⟩119889120573
119895
119904 119905 ge 0
(206)
On the other hand since 119883 is a local weak solution to (176)with initial condition ℎ and lifetime 120591 we have P-almostsurely for all 119905 ge 0 the identity
⟨120585 119883119905and120591
⟩ = ⟨120585 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904)⟩ 119889120573
119895
119904
(207)
Combining (206) and (207) yields up to indistinguishability
119861 + 119872 = 0 (208)
where the processes 119861 and 119872 are defined as
119861119905= int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩) 119889119904
119905 ge 0
119872119905= sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩ 119889120573
119895
119904
119905 ge 0
(209)
The process 119861 + 119872 is a continuous semimartingale withcanonical decomposition (208) Since the canonical decom-position of a continuous semimartingale is unique up toindistinguishability we deduce that 119861 = 119872 = 0 up toindistinguishability Using the Ito isometry (39) we obtain P-almost surely
int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩
+ ⟨120585 120572 (119883119904) minus 119863120601 (119884
showing that119883 is a local strong solution to (44) with lifetime120591
(3)rArr(1) This implication is a direct consequence ofProposition 29
The results from this section are closely related to the exis-tence of finite dimensional realizations that is the existenceof invariantmanifolds for each starting point ℎ
0 andwe point
out the papers [17ndash22] regarding this topic Furthermore wemention thatTheorem 68 has been extended in [23] to SPDEswith jumps
Acknowledgments
The author is grateful to Daniel Gaigall Georg GrafendorferFlorian Modler and Thomas Salfeld for the valuable com-ments and discussions
References
[1] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions CambridgeUniversity Press CambridgeUK 1992
[2] C Prevot and M Rockner A Concise Course on StochasticPartial Differential Equations Springer Berlin Germany 2007
[3] L Gawarecki and V Mandrekar Stochastic Differential Equa-tions in Infinite Dimensions with Applications to StochasticPartial Differential Equations Springer Berlin Germany 2011
[4] S Peszat and J Zabczyk Stochastic Partial Differential Equationswith Levy Noise Cambridge University Press Cambridge UK2007
[5] S Albeverio V Mandrekar and B Rudiger ldquoExistence of mildsolutions for stochastic differential equations and semilinearequations with non-Gaussian Levy noiserdquo Stochastic Processesand Their Applications vol 119 no 3 pp 835ndash863 2009
[6] D Filipovic S Tappe and J Teichmann ldquoJump-diffusions inHilbert spaces existence stability and numericsrdquo Stochasticsvol 82 no 5 pp 475ndash520 2010
[7] C Marinelli C Prevot and M Rockner ldquoRegular dependenceon initial data for stochastic evolution equations with multi-plicative Poisson noiserdquo Journal of Functional Analysis vol 258no 2 pp 616ndash649 2010
[8] S Tappe ldquoSome refinements of existence results for SPDEsdriven by Wiener processes and Poisson random measuresrdquoInternational Journal of Stochastic Analysis vol 2012 Article ID236327 24 pages 2012
[9] T Nakayama ldquoSupport theorem for mild solutions of SDErsquos inHilbert spacesrdquo Journal of Mathematical Sciences the Universityof Tokyo vol 11 no 3 pp 245ndash311 2004
[10] T Nakayama ldquoViability theorem for SPDErsquos including HJMframeworkrdquo Journal of Mathematical Sciences the University ofTokyo vol 11 no 3 pp 313ndash324 2004
[11] W Rudin Functional Analysis McGraw-Hill New York NYUSA 2nd edition 1991
[12] DWerner Funktionalanalysis Springer Berlin Germany 2007[13] C D Aliprantis and K C Border Infinite Dimensional Analysis
Springer Berlin Germany 3rd edition 2006[14] S N Ethier and T G KurtzMarkov Processes Characterization
andConvergenceWiley Series in Probability and Statistics JohnWiley amp Sons New Jersey NJ USA 1986
[15] V Mandrekar and B Rudiger ldquoExistence and uniqueness ofpath wise solutions for stochastic integral equations driven byLevy noise on separable Banach spacesrdquo Stochastics vol 78 no4 pp 189ndash212 2006
[16] D Filipovic Consistency Problems for Heath-Jarrow-MortonInterest Rate Models Springer Berlin Germany 2001
[17] T Bjork and L Svensson ldquoOn the existence of finite-dimensional realizations for nonlinear forward rate modelsrdquoMathematical Finance vol 11 no 2 pp 205ndash243 2001
[18] T Bjork and C Landen ldquoOn the construction of finite dimen-sional realizations for nonlinear forward rate modelsrdquo Financeand Stochastics vol 6 no 3 pp 303ndash331 2002
[19] D Filipovic and J Teichmann ldquoExistence of invariantmanifoldsfor stochastic equations in infinite dimensionrdquo Journal ofFunctional Analysis vol 197 no 2 pp 398ndash432 2003
[20] D Filipovic and J Teichmann ldquoOn the geometry of the termstructure of interest ratesrdquo Proceedings of The Royal Society ofLondon A vol 460 no 2041 pp 129ndash167 2004
[21] S Tappe ldquoAn alternative approach on the existence of affinerealizations for HJM term structure modelsrdquo Proceedings of
International Journal of Stochastic Analysis 25
TheRoyal Society of LondonA vol 466 no 2122 pp 3033ndash30602010
[22] S Tappe ldquoExistence of affine realizations for Levy term struc-ture modelsrdquo Proceedings of The Royal Society of London A vol468 no 2147 pp 3685ndash3704 2012
[23] D Filipovic S Tappe and J Teichmann ldquoInvariant manifoldswith boundary for jump-diffusionsrdquo httparxivorgabs12021076
Defining the R119898-valued process 119884 = ⟨120577 119883⟩ we have P-almost surely
119884119905and120591
isin 119881 forall119905 ge 0 (203)
Moreover since 119883 is a weak solution to (176) with initialcondition ℎ setting 119910 = ⟨120577 ℎ⟩ isin 119881 we have P-almost surely
119884119905and120591
= ⟨120577 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120577 119883119904⟩ + ⟨120577 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120577 120590119895
(119883119904)⟩ 119889120573
119895
119904
= ⟨120577 ℎ⟩ + int
119905and120591
0
120572120601120577
(⟨120577 119883119904⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(⟨120577 119883
119904⟩) 119889120573
119895
119904
= 119910 + int
119905and120591
0
120572120601120577
(119884119904) 119889119904
+ sum
119895isinN
int
119905and120591
0
120590119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(204)
showing that 119884 is a local strong solution to (192) with initialcondition 119910 By Itorsquos formula (Theorem 26) we obtain P-almost surely
119883119905and120591
= 120601 (119884119905and120591
)
= ℎ + int
119905and120591
0
(119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904)))119889119904
+ sum
119895isinN
int
119905and120591
0
119863120601 (119884119904) 120590
119895
120601120577(119884
119904) 119889120573
119895
119904 119905 ge 0
(205)
Now let 120585 isin D(119860lowast
) be arbitrary Then we have P-almostsurely
⟨120585 119883119905and120591
⟩
= ⟨120585 ℎ⟩
+ int
119905and120591
0
⟨120585119863120601 (119884119904) 120572
120601120577(119884
119904)
+1
2sum
119895isinN
1198632
120601 (119884119904) (120590
119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 119863120601 (119884119904) 120590
119895
120601120577(119884
119904)⟩119889120573
119895
119904 119905 ge 0
(206)
On the other hand since 119883 is a local weak solution to (176)with initial condition ℎ and lifetime 120591 we have P-almostsurely for all 119905 ge 0 the identity
⟨120585 119883119905and120591
⟩ = ⟨120585 ℎ⟩ + int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904)⟩) 119889119904
+ sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904)⟩ 119889120573
119895
119904
(207)
Combining (206) and (207) yields up to indistinguishability
119861 + 119872 = 0 (208)
where the processes 119861 and 119872 are defined as
119861119905= int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩ + ⟨120585 120572 (119883
119904) minus 119863120601 (119884
119904) 120572
120601120577(119884
119904)
minus1
2sum
119895isinN
1198632
120601 (119884119904)
times (120590119895
120601120577(119884
119904) 120590
119895
120601120577(119884
119904))⟩) 119889119904
119905 ge 0
119872119905= sum
119895isinN
int
119905and120591
0
⟨120585 120590119895
(119883119904) minus 119863120601 (119884
119904) 120590
119895
120601120577(119884
119904)⟩ 119889120573
119895
119904
119905 ge 0
(209)
The process 119861 + 119872 is a continuous semimartingale withcanonical decomposition (208) Since the canonical decom-position of a continuous semimartingale is unique up toindistinguishability we deduce that 119861 = 119872 = 0 up toindistinguishability Using the Ito isometry (39) we obtain P-almost surely
int
119905and120591
0
(⟨119860lowast
120585 119883119904⟩
+ ⟨120585 120572 (119883119904) minus 119863120601 (119884
showing that119883 is a local strong solution to (44) with lifetime120591
(3)rArr(1) This implication is a direct consequence ofProposition 29
The results from this section are closely related to the exis-tence of finite dimensional realizations that is the existenceof invariantmanifolds for each starting point ℎ
0 andwe point
out the papers [17ndash22] regarding this topic Furthermore wemention thatTheorem 68 has been extended in [23] to SPDEswith jumps
Acknowledgments
The author is grateful to Daniel Gaigall Georg GrafendorferFlorian Modler and Thomas Salfeld for the valuable com-ments and discussions
References
[1] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions CambridgeUniversity Press CambridgeUK 1992
[2] C Prevot and M Rockner A Concise Course on StochasticPartial Differential Equations Springer Berlin Germany 2007
[3] L Gawarecki and V Mandrekar Stochastic Differential Equa-tions in Infinite Dimensions with Applications to StochasticPartial Differential Equations Springer Berlin Germany 2011
[4] S Peszat and J Zabczyk Stochastic Partial Differential Equationswith Levy Noise Cambridge University Press Cambridge UK2007
[5] S Albeverio V Mandrekar and B Rudiger ldquoExistence of mildsolutions for stochastic differential equations and semilinearequations with non-Gaussian Levy noiserdquo Stochastic Processesand Their Applications vol 119 no 3 pp 835ndash863 2009
[6] D Filipovic S Tappe and J Teichmann ldquoJump-diffusions inHilbert spaces existence stability and numericsrdquo Stochasticsvol 82 no 5 pp 475ndash520 2010
[7] C Marinelli C Prevot and M Rockner ldquoRegular dependenceon initial data for stochastic evolution equations with multi-plicative Poisson noiserdquo Journal of Functional Analysis vol 258no 2 pp 616ndash649 2010
[8] S Tappe ldquoSome refinements of existence results for SPDEsdriven by Wiener processes and Poisson random measuresrdquoInternational Journal of Stochastic Analysis vol 2012 Article ID236327 24 pages 2012
[9] T Nakayama ldquoSupport theorem for mild solutions of SDErsquos inHilbert spacesrdquo Journal of Mathematical Sciences the Universityof Tokyo vol 11 no 3 pp 245ndash311 2004
[10] T Nakayama ldquoViability theorem for SPDErsquos including HJMframeworkrdquo Journal of Mathematical Sciences the University ofTokyo vol 11 no 3 pp 313ndash324 2004
[11] W Rudin Functional Analysis McGraw-Hill New York NYUSA 2nd edition 1991
[12] DWerner Funktionalanalysis Springer Berlin Germany 2007[13] C D Aliprantis and K C Border Infinite Dimensional Analysis
Springer Berlin Germany 3rd edition 2006[14] S N Ethier and T G KurtzMarkov Processes Characterization
andConvergenceWiley Series in Probability and Statistics JohnWiley amp Sons New Jersey NJ USA 1986
[15] V Mandrekar and B Rudiger ldquoExistence and uniqueness ofpath wise solutions for stochastic integral equations driven byLevy noise on separable Banach spacesrdquo Stochastics vol 78 no4 pp 189ndash212 2006
[16] D Filipovic Consistency Problems for Heath-Jarrow-MortonInterest Rate Models Springer Berlin Germany 2001
[17] T Bjork and L Svensson ldquoOn the existence of finite-dimensional realizations for nonlinear forward rate modelsrdquoMathematical Finance vol 11 no 2 pp 205ndash243 2001
[18] T Bjork and C Landen ldquoOn the construction of finite dimen-sional realizations for nonlinear forward rate modelsrdquo Financeand Stochastics vol 6 no 3 pp 303ndash331 2002
[19] D Filipovic and J Teichmann ldquoExistence of invariantmanifoldsfor stochastic equations in infinite dimensionrdquo Journal ofFunctional Analysis vol 197 no 2 pp 398ndash432 2003
[20] D Filipovic and J Teichmann ldquoOn the geometry of the termstructure of interest ratesrdquo Proceedings of The Royal Society ofLondon A vol 460 no 2041 pp 129ndash167 2004
[21] S Tappe ldquoAn alternative approach on the existence of affinerealizations for HJM term structure modelsrdquo Proceedings of
International Journal of Stochastic Analysis 25
TheRoyal Society of LondonA vol 466 no 2122 pp 3033ndash30602010
[22] S Tappe ldquoExistence of affine realizations for Levy term struc-ture modelsrdquo Proceedings of The Royal Society of London A vol468 no 2147 pp 3685ndash3704 2012
[23] D Filipovic S Tappe and J Teichmann ldquoInvariant manifoldswith boundary for jump-diffusionsrdquo httparxivorgabs12021076
showing that119883 is a local strong solution to (44) with lifetime120591
(3)rArr(1) This implication is a direct consequence ofProposition 29
The results from this section are closely related to the exis-tence of finite dimensional realizations that is the existenceof invariantmanifolds for each starting point ℎ
0 andwe point
out the papers [17ndash22] regarding this topic Furthermore wemention thatTheorem 68 has been extended in [23] to SPDEswith jumps
Acknowledgments
The author is grateful to Daniel Gaigall Georg GrafendorferFlorian Modler and Thomas Salfeld for the valuable com-ments and discussions
References
[1] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions CambridgeUniversity Press CambridgeUK 1992
[2] C Prevot and M Rockner A Concise Course on StochasticPartial Differential Equations Springer Berlin Germany 2007
[3] L Gawarecki and V Mandrekar Stochastic Differential Equa-tions in Infinite Dimensions with Applications to StochasticPartial Differential Equations Springer Berlin Germany 2011
[4] S Peszat and J Zabczyk Stochastic Partial Differential Equationswith Levy Noise Cambridge University Press Cambridge UK2007
[5] S Albeverio V Mandrekar and B Rudiger ldquoExistence of mildsolutions for stochastic differential equations and semilinearequations with non-Gaussian Levy noiserdquo Stochastic Processesand Their Applications vol 119 no 3 pp 835ndash863 2009
[6] D Filipovic S Tappe and J Teichmann ldquoJump-diffusions inHilbert spaces existence stability and numericsrdquo Stochasticsvol 82 no 5 pp 475ndash520 2010
[7] C Marinelli C Prevot and M Rockner ldquoRegular dependenceon initial data for stochastic evolution equations with multi-plicative Poisson noiserdquo Journal of Functional Analysis vol 258no 2 pp 616ndash649 2010
[8] S Tappe ldquoSome refinements of existence results for SPDEsdriven by Wiener processes and Poisson random measuresrdquoInternational Journal of Stochastic Analysis vol 2012 Article ID236327 24 pages 2012
[9] T Nakayama ldquoSupport theorem for mild solutions of SDErsquos inHilbert spacesrdquo Journal of Mathematical Sciences the Universityof Tokyo vol 11 no 3 pp 245ndash311 2004
[10] T Nakayama ldquoViability theorem for SPDErsquos including HJMframeworkrdquo Journal of Mathematical Sciences the University ofTokyo vol 11 no 3 pp 313ndash324 2004
[11] W Rudin Functional Analysis McGraw-Hill New York NYUSA 2nd edition 1991
[12] DWerner Funktionalanalysis Springer Berlin Germany 2007[13] C D Aliprantis and K C Border Infinite Dimensional Analysis
Springer Berlin Germany 3rd edition 2006[14] S N Ethier and T G KurtzMarkov Processes Characterization
andConvergenceWiley Series in Probability and Statistics JohnWiley amp Sons New Jersey NJ USA 1986
[15] V Mandrekar and B Rudiger ldquoExistence and uniqueness ofpath wise solutions for stochastic integral equations driven byLevy noise on separable Banach spacesrdquo Stochastics vol 78 no4 pp 189ndash212 2006
[16] D Filipovic Consistency Problems for Heath-Jarrow-MortonInterest Rate Models Springer Berlin Germany 2001
[17] T Bjork and L Svensson ldquoOn the existence of finite-dimensional realizations for nonlinear forward rate modelsrdquoMathematical Finance vol 11 no 2 pp 205ndash243 2001
[18] T Bjork and C Landen ldquoOn the construction of finite dimen-sional realizations for nonlinear forward rate modelsrdquo Financeand Stochastics vol 6 no 3 pp 303ndash331 2002
[19] D Filipovic and J Teichmann ldquoExistence of invariantmanifoldsfor stochastic equations in infinite dimensionrdquo Journal ofFunctional Analysis vol 197 no 2 pp 398ndash432 2003
[20] D Filipovic and J Teichmann ldquoOn the geometry of the termstructure of interest ratesrdquo Proceedings of The Royal Society ofLondon A vol 460 no 2041 pp 129ndash167 2004
[21] S Tappe ldquoAn alternative approach on the existence of affinerealizations for HJM term structure modelsrdquo Proceedings of
International Journal of Stochastic Analysis 25
TheRoyal Society of LondonA vol 466 no 2122 pp 3033ndash30602010
[22] S Tappe ldquoExistence of affine realizations for Levy term struc-ture modelsrdquo Proceedings of The Royal Society of London A vol468 no 2147 pp 3685ndash3704 2012
[23] D Filipovic S Tappe and J Teichmann ldquoInvariant manifoldswith boundary for jump-diffusionsrdquo httparxivorgabs12021076
showing that119883 is a local strong solution to (44) with lifetime120591
(3)rArr(1) This implication is a direct consequence ofProposition 29
The results from this section are closely related to the exis-tence of finite dimensional realizations that is the existenceof invariantmanifolds for each starting point ℎ
0 andwe point
out the papers [17ndash22] regarding this topic Furthermore wemention thatTheorem 68 has been extended in [23] to SPDEswith jumps
Acknowledgments
The author is grateful to Daniel Gaigall Georg GrafendorferFlorian Modler and Thomas Salfeld for the valuable com-ments and discussions
References
[1] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions CambridgeUniversity Press CambridgeUK 1992
[2] C Prevot and M Rockner A Concise Course on StochasticPartial Differential Equations Springer Berlin Germany 2007
[3] L Gawarecki and V Mandrekar Stochastic Differential Equa-tions in Infinite Dimensions with Applications to StochasticPartial Differential Equations Springer Berlin Germany 2011
[4] S Peszat and J Zabczyk Stochastic Partial Differential Equationswith Levy Noise Cambridge University Press Cambridge UK2007
[5] S Albeverio V Mandrekar and B Rudiger ldquoExistence of mildsolutions for stochastic differential equations and semilinearequations with non-Gaussian Levy noiserdquo Stochastic Processesand Their Applications vol 119 no 3 pp 835ndash863 2009
[6] D Filipovic S Tappe and J Teichmann ldquoJump-diffusions inHilbert spaces existence stability and numericsrdquo Stochasticsvol 82 no 5 pp 475ndash520 2010
[7] C Marinelli C Prevot and M Rockner ldquoRegular dependenceon initial data for stochastic evolution equations with multi-plicative Poisson noiserdquo Journal of Functional Analysis vol 258no 2 pp 616ndash649 2010
[8] S Tappe ldquoSome refinements of existence results for SPDEsdriven by Wiener processes and Poisson random measuresrdquoInternational Journal of Stochastic Analysis vol 2012 Article ID236327 24 pages 2012
[9] T Nakayama ldquoSupport theorem for mild solutions of SDErsquos inHilbert spacesrdquo Journal of Mathematical Sciences the Universityof Tokyo vol 11 no 3 pp 245ndash311 2004
[10] T Nakayama ldquoViability theorem for SPDErsquos including HJMframeworkrdquo Journal of Mathematical Sciences the University ofTokyo vol 11 no 3 pp 313ndash324 2004
[11] W Rudin Functional Analysis McGraw-Hill New York NYUSA 2nd edition 1991
[12] DWerner Funktionalanalysis Springer Berlin Germany 2007[13] C D Aliprantis and K C Border Infinite Dimensional Analysis
Springer Berlin Germany 3rd edition 2006[14] S N Ethier and T G KurtzMarkov Processes Characterization
andConvergenceWiley Series in Probability and Statistics JohnWiley amp Sons New Jersey NJ USA 1986
[15] V Mandrekar and B Rudiger ldquoExistence and uniqueness ofpath wise solutions for stochastic integral equations driven byLevy noise on separable Banach spacesrdquo Stochastics vol 78 no4 pp 189ndash212 2006
[16] D Filipovic Consistency Problems for Heath-Jarrow-MortonInterest Rate Models Springer Berlin Germany 2001
[17] T Bjork and L Svensson ldquoOn the existence of finite-dimensional realizations for nonlinear forward rate modelsrdquoMathematical Finance vol 11 no 2 pp 205ndash243 2001
[18] T Bjork and C Landen ldquoOn the construction of finite dimen-sional realizations for nonlinear forward rate modelsrdquo Financeand Stochastics vol 6 no 3 pp 303ndash331 2002
[19] D Filipovic and J Teichmann ldquoExistence of invariantmanifoldsfor stochastic equations in infinite dimensionrdquo Journal ofFunctional Analysis vol 197 no 2 pp 398ndash432 2003
[20] D Filipovic and J Teichmann ldquoOn the geometry of the termstructure of interest ratesrdquo Proceedings of The Royal Society ofLondon A vol 460 no 2041 pp 129ndash167 2004
[21] S Tappe ldquoAn alternative approach on the existence of affinerealizations for HJM term structure modelsrdquo Proceedings of
International Journal of Stochastic Analysis 25
TheRoyal Society of LondonA vol 466 no 2122 pp 3033ndash30602010
[22] S Tappe ldquoExistence of affine realizations for Levy term struc-ture modelsrdquo Proceedings of The Royal Society of London A vol468 no 2147 pp 3685ndash3704 2012
[23] D Filipovic S Tappe and J Teichmann ldquoInvariant manifoldswith boundary for jump-diffusionsrdquo httparxivorgabs12021076
TheRoyal Society of LondonA vol 466 no 2122 pp 3033ndash30602010
[22] S Tappe ldquoExistence of affine realizations for Levy term struc-ture modelsrdquo Proceedings of The Royal Society of London A vol468 no 2147 pp 3685ndash3704 2012
[23] D Filipovic S Tappe and J Teichmann ldquoInvariant manifoldswith boundary for jump-diffusionsrdquo httparxivorgabs12021076
![stochastic partial differential equations · methods. SofarGalerkinmethodsaremainly used for partial differential equations(cf. [36, 13, 35]) but first applications to stochastic](https://static.documents.pub/doc/80x56/5edcc847ad6a402d666799c0/stochastic-partial-differential-equations-methods-sofargalerkinmethodsaremainly.jpg)
