Stochastic Evolution Equations in inﬁnite dimension with...

Stochastic Evolution Equations in infinite

dimension with Applications to Term Structure

Problems

Josef Teichmann

Lecture Notes from lectures at CREST (Paris 2003), RTN-Workshop (Roscoff 2003), the MPILeipzig (Leipzig 2005) and the RICAM (Linz 2005)

Abstract. We provide an introduction to the mathematics of stochastic mod-

eling of term structure problems (like interest rate term structures, or recentlyvariance swap models). We then discuss geometric properties of time evolu-

tions such as finite dimensional realizations of the respective term structure

equations, or hypo-ellipticity. We mainly work with SDEs driven by (finitely orinfinitely many) Brownian motions. Stochastic Calculus of Variations (Malli-

avin Calculus) will be introduced to analyse law properties of several resulting

processes.The lecture notes decompose into an introduction to HJM-theory, the

question of finite dimensional realizations and its solution with Frobenius The-

ory. Finally methods from Malliavin Calculus are applied to investigate qual-itative properties of interest rate evolutions.

Contents

Introduction v1. Preface v

Chapter 1. Interest Rate Theory and Stochastic Evolution equations 11. Stochastic Differential equations with values in Hilbert spaces 12. Basic Notions of Interest Rate Theory 93. Classical Interest Rate models 104. HJM approach 135. The HJM equation as SDE with values in a Hilbert space 14

Chapter 2. Finite dimensional Realizations 191. Locally invariant submanifolds 192. Montgommery-Zippin Results on Semiflows 223. M ⊂ D(A∞) as sub-manifold 27

Chapter 3. Frobenius Theory 291. Geometric and Analytic Methods 292. Analysis 303. Geometry 334. Synthesis 355. Examples 37

Chapter 4. Malliavin Calculus in infinite dimensions 431. Gaussian Spaces and Malliavin Derivatives 432. The Skorohod Integral and Partial Integration 483. Existence of Densities 494. Hormander’s Theorem in finite dimensions 535. Hormander’s Theorem in infinite dimensions 61

Appendix A. The Wiener Chaos expansion 71

Appendix B. Cameron-Martin Space and Girsanov’s Theorem 83

Appendix. Bibliography 89

iii

Introduction

The lecture notes grew out of courses given in February 2003 in at CRESTin Paris, in May 2003 at a meeting of the European Union RTN in Roscoff, inApril 2005 at the MPI in Leipzig and in May 2005 at the RICAM in Linz. Thelectures in Paris, Roscoff and Leipzig covered the first three chapters, the lec-ture in Linz concentrated on the fourth chapter. I am grateful for the opportu-nity of giving lectures on theses topics to the organizers, in particular to HeinzEngl (RICAM Linz), Rudiger Frey (Universitat Leipzig), Elyees Jouini (UniversiteParis Dauphine), Marc Quincampoix (Universite Brest, coordinator of the Euro-pean Research Training Network HPRN-CT-2002-00281), Jorn Sass (RICAM Linz),Thorsten Schmid (Universitat Leipzig), Nizar Touzi (CREST Paris). These notescover the topics of the respective lectures and include additional material, whichwas not presented during the lectures. The material mainly stems from joint re-search articles with Fabrice Baudoin and Damir Filipovic and was inspired by worksof Tomas Bjork, Damir Filipovic and Lars Svensson.

1. Preface

I start this introductory chapter by introducing and solving several questionsin finite dimensional stochastic analysis. We shall (first) deal with stochastic dif-ferential equations in finite dimensions.

Definition 1. A map V : U ⊂ RN → RN is called vector-field on the open setU . The vector field V is called C∞-bounded if all derivatives dαV are bounded onU for multi-indices α ∈ NN \(0). We consider vector-fields as maps, but we shallsynonymously regard them as first order differential operators on U , i.e. given areal-valued smooth function f ∈ C∞(U),

V f(x) := df(x) · V (x)

for x ∈ U . Here and in the sequel d denotes the derivative of (differentiable) maps.

Given smooth vector fields V, V1, . . . , Vd : U → RN and a d-dimensional Brow-nian motion (Wt)t≥0 on a filtered probability space (Ω,F , P, (Ft)t≥0) with usualconditions, we consider the stochastic differential equation

dXxt = V (Xx

t )dt+d∑

i=1

Vi(Xxt )dW i

t ,

Xx0 = x.

The Picard-Lindelof approach guarantees local solvability of this equations, i.e.there exists a continuous, adapted process (Xx

t )t≥0 and a family of strictly positive

v

vi INTRODUCTION

stopping time τx > 0, such that

(0.1) Xxt = x+

∫ t

0

V (Xxs )ds+

d∑i=1

∫ t

0

Vi(Xxs )dW i

s

for 0 ≤ t ≤ τx. Furthermore

E(∫ t

0

||Xxs ||2ds) <∞.

Given two such processes with two families of stopping times, then on the commontime intervals the processes are indistinguishable.

Remark 1. Due to our convention that vector fields also represent first orderdifferential operators, there is a particularly simple way to write the generator ofthe above diffusion. We define

V0(x) := V (x)− 12

d∑i=1

dVi(x) · Vi(x)

for x ∈ U , hence the generator is the second-order operator

A = V0 +12

d∑i=1

(Vi)2.

Here the square means composition of differential operators.

We are interested in the following questions:• Do there exist locally invariant subsets M ⊂ U (sub-manifolds of U) for

this time evolution, i.e. that for any x ∈M there is a stopping time ηx > 0such that Xx

t ∈ M for 0 ≤ t ≤ ηx ∧ τx. What are minimal dimensionsof such objects? What is the possible shape? Can one reduce – by anintelligent transformation – the stochastic evolution in an N -dimensionalstate space to a stochastic evolution in a possibly lower dimensional statespace?

• Assume global solvability (i.e. τx = ∞) . If there is no lower dimensionalinvariant subset M , does the law of Xx

t admit a density? Can one easilycalculate the density and how to apply it to ”practical” questions. Whatcan we tell about the shape of the densities?

• Can similar questions be solved if one allows jumps, additionally?These questions are solved by the following two main theorems, which admit

already in finite dimensions a remarkable insight ”into” the diffusion.

Definition 2 (sub-manifold with boundary). Let U ⊂ RN be an open subset.M ⊂ U is called sub-manifold of U if for any x ∈ M we are given a sub-manifoldchart, i.e. there is a diffeomorphism ψ : U(x) ⊂ U → V (ψ(x)) ⊂ RN with U(x) anopen neighborhood of x in U , and V (ψ(x)) an open neighborhood of ψ(x) in RN ,such that

ψ(U(x) ∩M) = W (pr1(ψ(x))× pr2(ψ(x).with W (pr1(ψ(x)) an open neighborhood of pr1(ψ(x)) in Rm for some m ≥ 0. Herepri, i = 1, 2 denote the projections from RN on Rm and RN−m. The number m iscalled the local dimension of M and is obviously constant on connected components.A sub-manifold with boundary of U is defined similarly, except that we replace

1. PREFACE vii

Rm by Rm≥0, i.e. the set of vectors with last component non-negative. The locally

(m − 1)-dimensional set of points which are mapped in one (and then all sub-manifold charts) to 0 in the last component is called the boundary of M .

Definition 3 (tangent space). Let U ⊂ RN be an open subset and M ⊂ Usub-manifold with boundary of U . The tangent space TxM of M at x ∈M is definedas linear hull of tangent directions at x ∈ M . A smooth curve c : R → U is calledtangent at x ∈M if c(t) ∈M for all t ∈ R, if c(0) = x and d

dt |t=0pr2(ψ(c(t))) = 0.A tangent direction is then given by c′(0) ∈ RN . The inward-pointing tangentvectors at x ∈ TxM are those, which lie in the cone generated by tangent directions.We denote this space by (TxM)≥0.

Definition 4. Given equation (0.1) and a sub-manifold with boundary M ofU , we call M locally invariant with respect to the solution of equation (0.1) if forall x ∈M there exists a strictly positive stopping time ηx such that Xx

t ∈M for all0 ≤ t ≤ τx ∧ ηx.

Theorem 1. Given equation (0.1) and a sub-manifold with boundary M of U .Then M is locally invariant if and only if

Vi(x) ∈ TxM

for i = 1, . . . , d and x ∈M \ ∂M and

Vi(x) ∈ Tx∂M

V0(x) ∈ (TxM)≥0

for x ∈ ∂M .

Remark 2. This intuitively clear from the support theorem, which tells thatthe closure of the points y(T )

dy(t) = V0(y(t))dt+d∑

i=1

Vi(y(t))dωit,

y(0) = x

for finite total variation curves ω : [0, T ] → Rd coincides with the support of XxT .

Due to the conditions of the previous Theorem we obtain that y(T ) ∈ M if thetangent conditions are satisfied, for any finite total variation curve ω.

Proof. For the proof we refer to chapter 2.

Hence we can precisely read from the above theorem by the described localcondition of being tangent, whether a sub-manifold is locally invariant, or not. Thefollowing Lemma explains, that – given tangent vector-fields V,W – we can easilygenerate new tangent ones by Lie-bracketing.

Lemma 1. Let M ⊂ U be a submanifold with boundary of and X,Y tangent,smooth vector-fields on U , i.e. X(x), Y (x) ∈ TxM for x ∈M . Then the Lie bracket

[X,Y ](x) := dX(x) · Y (x)− dY (x) ·X(x)

for x ∈ U , is also tangent, [X,Y ](x) ∈ TxM .

viii INTRODUCTION

Example 1. Take the following diffusion with N = 2 and d = 1,

dXt = −12Xtdt+

(0 1−1 0

)XtdWt,

then V0(x) = 0 and

V1(x) =(

0 1−1 0

)x

for x ∈ R2. Hence the diffusion leaves any circle with radius r ≥ 0 locally invariant.By compactness of the circle the local invariance is in fact a global one.

Example 2. Take the following diffusion with N = 3 and d = 2,

dX1t = dW 1

t

dX2t = dW 2

t

dX3t = X1

t dW2t −X2

t dW1t .

Then we obtain

V0(x) = 0, V1(x) =

10−x2

, V2(x) =

01x1

,

which generates at any point x ∈ R3 a 2-dimensional sub-space, but since

[V1, V2](x) =

00−2

,

we obtain that a locally invariant sub-manifold necessarily has dimension 3, henceit is necessarily open. Consequently there is no non-trivial locally invariant sub-manifold.

Frobenius Theorems answer the existence question, which lies implicitly in thoseassertions. Given a certain diffusion of the type (0.1), do there exist locally invariantsub-manifolds and what is there dimension, shape? Here we refer to chapter 3, weshall only cite here the relevant result:

Theorem 2. Given equation (0.1) and x ∈ U , then there exists an m-dimensionalmanifold with boundary M ⊂ U being locally invariant and containing a pointy ∈ M , for any y ∈ U(x), where U(x) is an open neighborhood of x in U , if andonly if the distribution generated by V0, . . . , Vd is m-dimensional on U(x).

Remark 3. The distribution generated by d + 1 vector-fields V0, . . . , Vd on Uis given by a set of vector spaces, for any x ∈ U , namely

〈V0(x), . . . , Vd(x), [Vi, Vj ](x), . . . 〉 .

This complete treatment of the question, whether there exists a locally invariantsub-manifold for a locally defined diffusion, leads to the question, what happensif we do not have a locally invariant sub-manifold: intuitively the diffusion fillsopen sets. Does this mean that the law of the random variable Xx

t is absolutelycontinuous with respect to Lebesgue measure? In fact the answer is yes - with”trivial” exceptions. The question, whether the law admits a density, or not, canbe rephrased by the following generic property for random variables (which will

1. PREFACE ix

play an important role in infinite dimensions), namely that for any projection pr :RN → Rm (m ≤ N), the random variable pr(Xx

t ) admits a density.These questions are generically answered by Malliavin Calculus: in the sequel I

shall describe the relevant results for this question quickly, more details can be foundin standard books on Malliavin Calculus. We shall consider stochastic differentialequations of the type

dXxt = V0(Xx

t )dt+d∑

i=1

Vi(Xxt ) dBi

t

on a stochastic basis (Ω,F , P ) with d-dimensional Brownian motion (W it )0≤t≤T,i=1,...,d

up to a finite time horizon T > 0. For the vector fields Vi we shall assume that thevector fields

Vi : RN → RN

are C∞-bounded, since we need global solutions.The condition that the distribution

(0.2) DLA(x) := 〈V1(x), . . . , Vd(x), [Vi, Vj ](x), . . . for i, j, · · · = 0, . . . , d〉

for has constant rank N at a point x ∈ RN is called Hormander’s condition at x.We shall apply usual notions of Malliavin calculus. The first variation J0→t(x)

denotes the derivative of (Xxs )0≤s≤t with respect to x, hence

dJ0→t(x) = dV0(Xxt )J0→t(x)dt+

d∑i=1

dVi(Xxt )J0→t(x) dW i

t

J0→0(x) = idn.

This is an almost surely invertible process and we obtain the representation of theMalliavin derivative

DksX

xt = J0→t(x)(J0→s(x))−1Vk(Xx

s )1[0,t](s)

for 0 ≤ s ≤ T , 1 ≤ k ≤ d. We have the fundamental partial integration formulad∑

k=1

E(∫ t

0

DksX

xt a

kds) = E(Xxt δ(a)

T ),

where (as)0≤s≤t := (ak0≤s≤t)k=1,...,d ∈ dom(δ) is a Skorohod integrable processes.

δ(a) is real valued random variable. Notice that if a is predictable, then

δ(a) =d∑

k=1

∫ t

0

aksdW

ks ,

hence the Skorohod integral coincides with the Ito integral on predictable processes.

Remark 4. We could equally calculate all quantities in the sequel on manifolds.

Remark 5. If we assume that the distribution DLA(x) has constant rank n ≤ Nin neighborhood of x0 (and if it contains V0 at each point), then we obtain thatE(f(Xx

t )) is differentiable in all directions at x ∈ RN if and only if n = N . Ifn < N we find immersed sub-manifolds such that the process remains on thosesubmanifolds almost surely and Hormander’s condition holds true for the consideredprocess in the manifold.

x INTRODUCTION

The following Theorem of Paul Malliavin clarifies – in a probabilistic way –the situation under Hormander’s condition (0.2), namely that Xx

t admits a densitywith respect to Lebesgue’s measure. We recognize that Hormander’s condition issimilar (except the drift term V0) to the Frobenius condition on the existence oflocally invariant sub-manifolds.

Theorem 3. Assume Hormander’s condition at x ∈ RN , then for p ≥ 1 thecovariance matrix is invertible and has finite p-norm. From this we easily conclude(see also the sequel) that Xx

t admits a density.

Due to importance for application we also consider the first derivative of thefunction x 7→ E(f(Xx

t )) for t > 0 and explicit formulas, which we obtain for thisobject. By Hormander’s theorem we know that this function is smooth for allbounded measurable functions f : RN → R. We first show general smoothness,however, our goal is to achieve a feasible formula for the corresponding weightfunctions.

Theorem 4. Fix t > 0, w ∈ RN and x ∈ RN . Under the above assumptionthere is weight π ∈ D∞ such that for all measurable functions f : RN → R theformula

d

dt|t=0E(f(Xx+tw)) = E(f(Xx

t )π)

holds true.

Proof. We apply standard arguments from Malliavin Calculus. Assume f isC1

b , thend

dt|t=0E(f(Xx+tw)) = E(df(Xx

t ) · J0→t(x) · w),

where df denotes the differential of f . If we are able to solve the equation

w =d∑

k=1

∫ t

0

J0→s(x)−1Vk(Xxs )ak

sds

with a Skorohod integrable strategy such that δ(a) ∈ D∞, then we have proved thetheorem. Assume that we have a solution satisfying the requirements, then

d

dt|t=0E(f(Xx+tw)) = E(df(Xx

t ) · J0→t(x) · w)

= E(df(Xxt ) · J0→t(x) ·

d∑k=1

∫ t

0

J0→s(x)−1Vk(Xxs )ak

sds)

= E(d∑

k=1

∫ t

0

df(Xxt ) · J0→t(x) · J0→s(x)−1Vk(Xx

s )aksds)

= E(d∑

k=1

∫ t

0

DksX

xt a

ksds)

= E(f(Xxt )δ(a)),

hence the result.

1. PREFACE xi

The above equation can be solved directly by the reduced covariance matrixCt(x), namely

Ct(x) :=d∑

k=1

∫ t

0

J0→s(x)−1Vk(Xxs )(J0→s(x)−1Vk(Xx

s ))T ds,

which is under Hormander’s condition 0.2 almost surely invertible with 1det(Ct(x)) ∈

Lp (see [30]). Hence

aks := (J0→s(x)−1Vk(Xx

s ))T (Ct(x))−1 · wfor 0 ≤ s ≤ t solves the above problem with an anticipative strategy, where theresulting Skorohod integral lies in D∞. Furthermore the outcome δ(a) dependssmoothly on x. Hence we can conclude by classical arguments that the aboveformula which is valid for C1

b functions is indeed true for all bounded measurablefunctions.

Corollary 1. Fix t > 0, w ∈ RN and x ∈ RN and take the assumptions ofTheorem 4. For two weights π and η we obtain

E(π|σ(Xxt )) = E(η|σ(Xx

t )).

In particular the random variable E(π|σ(Xxt )) itself is a weight. For any Skorohod

integrable process a, such that δ(a) is a weight, we obtain

E(df(Xxt ) · J0→t(x) · w) = E(df(Xx

t ) · J0→t(x) ·d∑

k=1

∫ t

0

J0→s(x)−1V k(Xxs )ak

sds)

for all f ∈ C1b . This is a necessary and sufficient condition for a Skorohod integrable

process a, such that δ(a) is a weight. Furthermore there is a unique predictableprocess a such that δ(a) = E(π|σ(Xx

t )).

Proof. Take the defining equations.

Hence local or global Hormander conditions clarify the situation, if no locallyinvariant sub-manifolds do exist. The picture, which we obtain in finite dimen-sions, can now be carried over to the infinite dimensional world, where we haveapplications from term structure problems in mind.

In interest rate theory the following questions is of particular importance:(1) Assume that you ”have” today’s prices of zero-coupon bonds, i.e. the

curve T 7→ P (0, T ). Is it possible to find an interest rate model such ex-actly the given curve is its initial value (under some regularity assumptionson T 7→ P (0, T )).

(2) Is it always possible to determine factors of the forward rate evolution,which correspond to economic quantities such as the short rate process?

To treat these two questions we have to reconsider the presentation of thetheory again. The ”initial value” T 7→ P (0, T ) is somehow hidden in the short rateprocess and it is a priori not clear which short rate processes lead to which ”initialvalues”. Modelling the forward rate directly would include the initial value fromthe beginning and possibly make the theory more coherent.

CHAPTER 1

Interest Rate Theory and Stochastic Evolutionequations

1. Stochastic Differential equations with values in Hilbert spaces

To introduce the relevant notions on separable Hilbert spaces, we shall in-troduce some notions on strongly continuous semigroups, which are basic for thesequel. In the fifties the fundaments of solvability of initial value problems werelaid down by connecting knowledge of spectral properties with the solvability ofAbstract Cauchy Problems. The old method of solving ordinary differential equa-tions by Laplace transforms awarded new merits. These fundaments were enrichedby additional assumptions on the considered Banach spaces as for example latticestructures or C∗-algebra-structures. In this sectionX,Y denote Banach spaces, T, Sdenote C0-semigroups of continuous linear operators. A semigroup homomorphismis understood to map the identity to the identity.

Definition 5. Let (A,D(A)) be a closed operator on a Banach space X, theAbstract Cauchy Problem (ACP ) associated to A with initial value f ∈ D(A) is thesolution of

u ∈ C1(R≥0, X)

u(0) = f and u(t) ∈ D(A) for t ≥ 0d

dtu(t) = Au(t)

Definition 6 (C0-semigroup). A strongly continuous semigroup of linear op-erators on a Banach space X is a semigroup homomorphism T : R≥0 → L(X)with

limt↓0

Ttx = x

These semigroups are often referred to as C0-semigroups.

Remark 6. T is a strongly continuous semigroup if and only if T : R≥0 →L(X)s is continuous, where the bounded linear operators carry the strong topology.By Banach-Steinhaus (uniform boundedness principle) there are constants M ≥ 1and ω ∈ R such that

||Tt|| ≤M exp(ωt) for t ≥ 0The smallest possible value of ω is called the growth bound ω(T ) and might be −∞,the formula

ω(T ) = limt↓0

ln ||Tt||t

is valid. A semigroup T is called bounded if ||Tt|| ≤ M for t ≥ 0 , T is calledcontraction semigroup if M = 1 is a possible choice.

1

2 1. INTEREST RATE THEORY AND STOCHASTIC EVOLUTION EQUATIONS

Definition 7. Let T be a strongly continuous semigroup, then the infinitesimalgenerator (A,D(A)) is defined in the following way:

D(A) = x ∈ X | limt↓0

Ttx− x

texists

Ax = limt↓0

Ttx− x

tfor x ∈ D(A)

Remark 7. Generically the operator (A,D(A)) is unbounded on the Banachspace X. The subtle relation between the infinitesimal generator and the globalobject, the strongly continuous semigroup, is the subject of Hille-Yosida-Theory.

Theorem 5. Let (A,D(A)) be the generator of a strongly continuous semi-group, then the following assertions are valid:

(1) If x ∈ D(A), then Ttx ∈ D(A) for t ≥ 0(2) The map t 7→ Ttx is differentiable if and only if x ∈ D(A):

d

dtTtx = ATtx = TtAx for x ∈ D(A)

(3) For x ∈ X and φ ∈ C∞(R≥0,R) the integral∫ t

0φ(s)Tsxds lies in D(A)

for any t ≥ 0 and

A

∫ t

0

φ(s)Tsxds = φ(t)Ttx− φ(0)x−∫ t

0

φ′(s)Tsxds

(4) The domain D(A) is dense in X and (A,D(A)) is a closed operator. Fur-thermore (D(An), ||.||+ ||A.||+ ...+ ||An.||) are Banach spaces with D(An)dense in X. The intersection ∩n≥1D(An) is dense in X and equipped withthe initial topology via An : ∩n≥1D(An) → X, n ∈ N a Frechet space.

(5) There is only one semigroup with infinitesimal generator A.

Proof. The first and second statement are clear by definition: The integralexists and for h < t we obtain

Th − id

h

∫ t

0

φ(s)Tsxds =1h

(∫ t+h

h

φ(s− h)Tsxds−∫ t

0

φ(s)Tsxds) =

= −∫ t+h

h

φ(s)− φ(s− h)h

Tsxds+1h

(∫ t+h

h

φ(s)Tsxds−∫ t

0

φ(s)Tsxds) =

= −∫ t+h

h

φ(s)− φ(s− h)h

Tsxds+1h

∫ t+h

t

φ(s)Tsxds−1h

∫ h

0

φ(s)Tsxds)h→0→

= −∫ t

0

φ′(s)Tsxds+ φ(t)Ttx− φ(0)x

which proves the third assertion. Taking a Dirac sequence φεε>0 right from zerowith support in R>0 we see that

∫ 1

0φε(s)Tsxds→ x for x ∈ X and

A

∫ 1

0

φε(s)Tsxds =∫ 1

0

φ′ε(s)Tsxds

Consequently all domains D(An) and their intersection are dense in X. Closednessof A will be proved directly. Ttx−x =

∫ t

0TtAxdt by 2. and if xm → x and Axm → y,

then by continuity Ttx− x =∫ t

0Ttydt, so x ∈ D(A) and y = Ax. So (D(A), ||.||+

||A.||) is a Banach space. Assume that (D(An), ||.||+ ||A.||+ ...+ ||An.||) is a Banach

1. STOCHASTIC DIFFERENTIAL EQUATIONS WITH VALUES IN HILBERT SPACES 3

space as inductive hypothesis. By 2. D(An) is Tt-invariant, we can restrict thesemigroup to D(An) and obtain a semigroup of bounded linear operators, whichis strongly continuous in the complete topology. The infinitesimal generator isA|D(An+1), which is closed by induction. So (D(An+1), ||.||+ ||A.||+ ...+ ||An+1.||)is a Banach space. This establishes by the way the completeness of the intersectionas a Frechet space.

Assume that there is another strongly continuous semigroup S with infinitesi-mal generator A, then

Ttx− Stx =∫ t

0

d

ds(St−sTsx)ds =

∫ t

0

(St−sTs)(Ax−Ax)ds = 0

for x ∈ D(A), consequently by continuity S = T .

Remark 8. The abstract Cauchy Problem associated to a closed operator A isuniquely solvable for every f ∈ D(A) if and only if (A1, D(A2)) is the infinitesi-mal generator of a strongly continuous semigroup T 1 on the Banach space E1 :=(D(A), ||.||+ ||A.||). A1f = Af for f ∈ D(A2). Assume unique solvability on D(A)of the abstract Cauchy problem associated to A. Then we can associate a stronglycontinuous semigroup T 1 of linear operators. We have to prove that T 1

t is con-tinuous and that the infinitesimal generator is (A1, D(A2)): We denote the uniqueC1-solution of (ACP ) with initial value f by u(., f). T 1

t f = u(t, f) for f ∈ D(A)and t ≥ 0. We investigate η : E1 → C([0, t], E1), f 7→ u(., f): Let fn → f andη(fn) → v be sequences, then u(s, fn) = fn +

∫ s

0Au(r, fn)dr → v(s) = f +

∫ s

0Avdr,

then v is continuously differentiable on [0, t]. Continuing by T 1 gives the desiredv(s) = u(s, f) by uniqueness on [0, t]. So η has closed graph and is consequentlycontinuous, evaluation at s ∈ [0, t] yields that T 1 is a strongly continuous semi-group of bounded linear operators. Denote the infinitesimal generator of T 1by B.First we show that T 1

t Af = AT 1t f for f ∈ D(A2). v(t) := f +

∫ t

0u(s,Af)ds,

then ddtv(t) = u(t, Af) = Af +

∫ t

0Au(r,Af)dr = Av(t), so v(t) = u(t, f) and

Au(t, f) = ddtv(t) = u(t, Af). The rest is given by stating the definitions.

Assume that there is a strongly continuous semigroup T 1 with infinitesimalgenerator A1, then unique solvability follows from the definitions directly.

The existence of a strongly continuous semigroup T with infinitesimal generatorA, a closed operator, is equivalent to unique solvability of (ACP ) on D(A) andρ(A) 6= ∅, because (λ−A)−1 : E → E1 is an equivalence between the two semigroupsif they exist in the sense of representation theory.

The relation between spectral properties of the infinitesimal generator and C0-semigroups is clarified by the Laplace transform. The existence of the Laplacetransform leads to several strong properties of the resolvent and these strong as-ymptotic properties are seen to be sufficient for the existence. For φλ(t) = exp(−λt)we see that for λ > ω the limit t→∞ exists, consequently we arrive at the formulaby 1.6.2.:

(λ−A)∫ ∞

0

exp(−λt)Ttxdt = x

for all x ∈ X and λ > ω. This formula is valid for complex Banach spaces, too. Inthis case we can assert the half-plane right from ω lies in the resolvent set, we willemphasize further properties of the resolvent set in the case of analytic semigroups.


Consequently the Laplace transform for λ > ω is the resolvent R(λ,A) :=(λ−A)−1 of the infinitesimal generator. If the resolvent set ρ(A) of a closed operatorA on a complex Banach space is not empty, then it is open and the resolvent isanalytic and satisfies the resolvent equation

R(λ,A)R(µ,A)(µ− λ) = R(λ,A)−R(µ,A)

for λ, µ ∈ ρ(A). Immediately this functional equation leads to

R(λ,A)(n) = (−1)nn!R(λ,A)n+1

On the other hand by n-fold differentiation of the Laplace transform one obtains∫ ∞

0

(−1)ntn exp(−λt)Ttxdt

The identity leads to the well-known condition:

R(λ,A)n+1x =∫ ∞

0

tn

n!exp(−λt)Ttxdt

for <λ > ω and x ∈ X , n ∈ N. The estimate of the right hand side via theexponential growth constants of T leads to the Hille-Yosida-condition by partialintegration:

||R(λ,A)n|| ≤ M

(<λ− ω)n

for <λ > ω and n ∈ N. Surprisingly this simple asymptotic condition on theresolvent is already sufficient for the existence of a strongly continuous semigroupwith infinitesimal generator A, which is the contents of the Feller-Hille-Miyadera-Philipps-Yosida-Theorem:

Theorem 6 (Hille-Yosida-Theorem). Let X be a Banach space and(A,D(A)) a densely defined closed operator, then the following assertions are equiv-alent:

(1) A is the infinitesimal generator of a strongly continuous semigroup.(2) There are constants M ≥ 1, ω > 0 such that ]ω,∞[⊂ ρ(A) and

||R(λ,A)n|| ≤ M

(λ− ω)n

for n ∈ N and λ > ω.

Proof. Necessity was already shown by the previous remarks, so we assume2.: We define Aλ := λAR(λ,A) = λ(λR(λ,A)− id), so a continuous operator. Forx ∈ D(A)

||AR(λ,A)x|| = ||R(λ,A)Ax|| ≤ M

λ− ω||Ax|| λ→∞→ 0

||AR(λ,A)|| = ||λR(λ,A)− id|| ≤ λM

λ− ω+ 1

and by denseness of D(A) we can conclude that AR(λ,A)x → 0 for x ∈ X, soλR(λ,A)x → x for x ∈ X. This means finally Aλx = λR(λ,A)Ax → Ax for


x ∈ D(A). Since Aλ is bounded the exponential exists and satisfies the followingestimate on ]2ω,∞[

|| exp(tAλ)|| ≤ exp(−λt)∑n∈N

tnλ2n

n!||R(λ,A)n||

≤ exp(−λt)∑n∈N

tnλ2n

n!M

(λ− ω)n

≤M exp(tωλ

λ− ω) ≤M exp(2ωt)

for t ≥ 0. Consequently we can hope for convergence if λ tends to infinity:

|| exp(tAλ)x− exp(tAµ)x|| ≤ ||∫ t

0

d

ds(exp((t− s)Aλ) exp(sAµ))xds||

≤ ||∫ t

0

exp((t− s)Aλ) exp(sAµ)(Aµx−Aλx)ds||

≤M2 exp(4ωt)t||Aµx−Aλx|| → 0

as λ, µ tend to infinity uniformly on bounded intervals for x ∈ D(A). By theexponential boundedness exp(tAλ)x converges uniformly on bounded intervals toTtx for all x ∈ X, where the limit Ttx is a continuous curve and Tt is a boundedlinear operator. The semigroup property is preserved by the limit, too. We haveto show, that the generator A′ of the strongly continuous semigroup T is A. Theformula exp(Aλt)x− x =

∫ t

0exp(Aλs)Aλxds tends to

Ttx− x =∫ t

0

TsAxds

as λ → ∞ for x ∈ D(A). Consequently x ∈ D(A′) and A′x = Ax. By theasymptotic properties of exp(Aλt) we know by application of the Laplace transformthat λ−A′ = λ−A are both one-to-one and onto X for λ > 2ω, so A = A′.

The Hille-Yosida Theorem is a special case of the more general Chernoff The-orem, which shall be stated without proof here:

Theorem 7. Let c : R≥0 → L(X) be a strongly continuous curve of continuous,linear operators. Assume that ||c(t)n|| ≤M for 0 ≤ t ≤ ε and n ≥ 0 with a constantM ≥ 1. Furthermore we assume that c(0) = id and that there is a densely definedoperator A : D(A) → X such that

limt→0

c(t)x− x

t= Ax

for x ∈ D(A), and there is λ0 > 0 such that (λ0 − A)D(A) is dense. Then theclosure of A is the generator of a strongly continuous semi-group T and

Tt = limn→∞

c(t

n)n

uniformly on compact sets in the strong topology.

Having the notion of semigroups we need to understand stochastic integrationwith values in Hilbert spaces, which is a trivial extension for finitely many Brownianmotions (see for instance [9] or [30]).


We define as usual progressively measurable processes with values in a Hilbertspace H (we do not need separability here) and denote the square integrable pro-cesses of this class by L2(R≥0 × Ω,Fp, dt⊗ P ;H), i.e. the set of

φ : R≥0 × Ω → R,which are measurable with respect to the σ-algebra Fp, the σ-algebra generated byB([0, t])⊗Ft for t ≥ 0 and square-integrable thereon. These are all maps such thatthe restriction φ1[0,t] lies in L2([0, t]× Ω,B([0, t])⊗Ft, dt⊗ P ) and

E(∫ ∞

0

||φ(s)||2) =∫

Ω

∫ ∞

0

||φ(s, ω)||2dsP (dω) <∞.

The subspace of simple predictable processes, i.e.

uj(t) =n−1∑i=0

F ji 1]ti,ti+1](t)

with F ji a Fti

-measurable and E(||F ji ||2) <∞ (hence F j

i ∈ L2(Ω,Fti, P ;H)), n ≥ 0

and 0 = t0 ≤ t1 ≤ ... ≤ tn, is denoted by E . On E we define the Ito-integral by

I(u) =∫ ∞

0

uj(t)dWt :=n−1∑i=0

F ji (W j

ti+1−W j

ti)

Theorem 8. The mapping I : E → L2(Ω,F , P ) is a well defined isometry andE(I(u)) = 0 for all u ∈ E, i.e.

E(〈I(u), I(v)〉) =d∑

j=1

E(∫ ∞

0

⟨uj(t), vj(t)

⟩dt)

Theorem 9. The vector space E is dense in L2(R≥0 × Ω,Fp, dt⊗ P ;H).

Proof. The proof is done by the following steps, where we replace R≥0 by[0, 1], we can furthermore assume H = R: First we can approximate u ∈ L2([0, 1]×Ω,Fp, dt⊗P ) by a bounded, progressively measurable process by cutting down. Thebounded progressively measurable process u′ can be approximated by continuousadapted process, which can be approximated by a simple predictable step processin E . Given a bounded, progressively measurable process u′ we define

u′h(t) :=1h

∫ t

t−h

u′(s)ds

for t, h ≥ 0 where we assume that u′(s) = 0 for s ≤ 0. By Lebesgue’s differentiationtheorem we obtain for a fixed ω ∈ Ω that for the trajectories u′h(t)(ω) → u′(t)(ω)holds almost everywhere in t ∈ [0, 1] as h→ 0. Consequently∫ 1

0

|u′h(s)(ω)− u′(s)(ω)|2ds→ 0

by dominated convergence as h → 0 for all ω ∈ Ω. Again by boundedness anddominated convergence we conclude that

E(∫ 1

0

|u′h(s)(ω)− u′(s)(ω)|2ds) → 0

which is by Fubini’s theorem the integral with respect to the product measure.Hence by continuity and adaptedness of the processes u′h we conclude the result.


We take the usual assumption on the stochastic basis (Ω,F , Q), the filtration(Ft)t≥0 and the d-dimensional Brownian motion (W 1

t , . . . ,Wdt )t≥0. In our setting

a square-integrable Ito process is given through

Xt = X0 +∫ t

0

bsds+d∑

i=1

∫ d

0

aisdW

is

with b integrable and ai square-integrable, progressively measurable H-valued pro-cesses. Ito’s formula reads as follows:

Theorem 10. Suppose X is an H-valued Ito process and ψ ∈ C2b (H;H). Then

dψ(Xt)(ait) and

Jt :=12

d∑i=1

d2ψ(Xt)(ait, a

it)

satisfy the square-integrability conditions for H-valued Ito integrals and

(ψ X)t = ψ(X0) +∫ t

0

(dψ(Xs)(bs) + Js)ds+

+d∑

i=1

∫ t

0

dψ(Xs)(ais)dW

is .

This theorem also has a local version for ψ ∈ C2(H;H).

For the general theory we investigate the following equation

(1.1) dφt = (Aφt + α(φt))dt+d∑

i=1

βi(φt)dW it ,

where α, β1, . . . , βd : U → H are locally Lipschitz vector fields on the open subsetU ⊂ H and A is the generator of a strongly continuous semigroup (Tt)t≥0 on H.

In the sequel we shall assume for a stochastic process with continuous paths(φt)t≥0 that up to the given stopping time τ > 0

P (∫ t∧τ

0

(||φs||+ ||α(φs)||+d∑

i=1

||βi(φs)||2)ds <∞) = 1.

Solutions of equation 1.1 are defined in a subtle way. A strong, continuoussolution of 1.1 with initial value φ∗ is a stochastic process with continuous paths(φt)t≥0 together with a strictly positive stopping time τ such that φt ∈ D(A) for0 ≤ t ≤ τ ,

P (∫ t∧τ

0

||Aφs||ds <∞) = 1

and

φt = φ∗ +∫ t

0

Aφsds+∫ t

0

α(φs)ds+d∑

i=1

∫ t

0

βi(φs)dW is

for 0 ≤ t ≤ τ .A mild, continuous solution of 1.1 with initial value φ∗is a stochastic process

with continuous paths (φt)t≥0 together with a strictly positive stopping time τ such


that

φt = Ttφ∗ +

∫ t

0

Tt−sα(φs)ds+d∑

i=1

∫ t

0

Tt−sβi(φs)dW is

for 0 ≤ t ≤ τ . Clearly every strong, continuous solution is a mild, continuoussolution by variation of constants.

Proposition 1. For any initial value φ∗ ∈ U there is a unique local mild, con-tinuous solution of equation 1.1 under the stated Lipschitz conditions on α, β1, . . . , βd.Given φ∗ ∈ U , we can choose an open neighborhood W of φ∗ in U and a stoppingtime τ , such that there exist mild, continuous solutions (ηt)t≥0 with initial valueη0 ∈W and associated stopping time τ . Furthermore

E( sup0≤t≤τ

||ηnt − ηt||2H) → 0

if ηn0 → η0 in H for all sequences (ηn

0 )n≥1in W , η0 ∈W .

Proof. For the proof and the notions of uniqueness, also for stochastic flowson Hilbert spaces, see [9].

For this theory a third notion of solution is particular importance, which is alsoequivalent to mild solutions. A weak, continuous solution of 1.1 with initial valueφ∗is a stochastic process with continuous paths (φt)t≥0 together with a strictlypositive stopping time τ such that

〈φt, ξ〉 = 〈φ∗, ξ〉+∫ t

0

〈φs, A′ξ〉 ds+

∫ t

0

〈α(φs), ξ〉 ds+d∑

i=1

∫ t

0

〈βi(φs), ξ〉 dW is

for 0 ≤ t ≤ τ for ξ ∈ D(A′). We shall prove the following theorem:

Theorem 11. Let (φt)t≥0 be a stochastic process with continuous paths andφ0 = φ∗ and τ a strictly positive stopping time. Assume that

E(d∑

i=1

∫ t∧τ

0

||βi(φs)||2ds) <∞,

then (φt)t≥0 is a weak solution up to τ of 1.1 if and only if it is a mild solution upto τ . In fact the integrability assumption is only needed if we want to conclude thata weak solution is mild. The other direction works in general.

Proof. First we assume (φt)t≥0to be a mild solution, then

〈φt, ξ〉 = 〈Ttφ∗, ξ〉+

∫ t

0

〈Tt−sα(φs), ξ〉 ds+d∑

i=1

∫ t

0

〈Tt−sβi(φs), ξ〉 dW is

2. BASIC NOTIONS OF INTEREST RATE THEORY 9

holds true for 0 ≤ t ≤ τ for ξ ∈ D(A′). Hence taking the stochastic differentialyields of the real-valued semimartingale in question we obtain

d 〈φt, ξ〉 = (〈Ttφ∗, A′ξ〉+ 〈α(φt), ξ〉+

∫ t

0

〈Tt−sα(φs), A′ξ〉 ds)dt+

+d∑

i=1

〈βi(φt), ξ〉 dW it +

d∑i=1

(∫ t

0

〈Tt−sβi(φs), A′ξ〉 dW is)dt

= 〈φt, A′ξ〉 dt+ 〈α(φt), ξ〉 dt+

d∑i=1

〈βi(φt), ξ〉 dW it

by definition of a mild solution and commutation of integrals with linear functionals.Given a weak solution (φt)t≥0 we can apply partial integration

d⟨φs, T

′t−sξ

⟩=⟨φs, A

′T ′t−sξ⟩ds+

⟨α(φs), T ′t−sξ

⟩ds+

d∑i=1

⟨βi(φs), T ′t−sξ

⟩dW i

s+

−⟨φs, A

′T ′t−sξ⟩ds

for ξ ∈ D(A′), which yields immediately by integration the property of mild solu-tions.

In order to formulate term structure problems with thus need to specify aHilbert space of relevant curves, where typically the shift semigroup is a stronglycontinuous semigroup (Musiela parametrization).

2. Basic Notions of Interest Rate Theory

In the sequel we shall apply the following basic notions of Interest Rate The-ory. We are dealing with a market of zero-coupon bonds, which are traded assetscharacterized by a maturity time T . We denote the price of a zero-coupon bondwith maturity T ≥ 0 at time t ≤ T by P (t, T ). We assume that we are given pricesfor all maturity times T ≥ 0 and for all time t ≤ T . This continuity assumptionis certainly strong, however, not stronger as in the Black-Scholes world. Prices areassumed to be positive quantities, i.e. P (t, T ) > 0 and to be normalized at t = T ,i.e. P (T, T ) = 1. Therefore P (t, T ) represents the discounting factor from timeT to t. We analyze the stochastic nature interest rate models: suppose we aregiven a (Ω,F , P ) with filtration (Ft)t≥0. We assume the filtration to satisfy theusual conditions. Prices of default-free zero coupon bonds P (t, T ) are modelled assemimartingales with continuous paths for 0 ≤ t ≤ T with respect to (Ft)0≤t≤T .The process (P (t, T ))0≤t≤T is positive almost surely, furthermore we assume thenormalization P (T, T ) = 1 almost surely. No arbitrage in this setting is usuallygiven by the following requirement, which we shall assume throughout: there ex-ists a probability measure Q and the (Ft)t≥0-adapted interest rate process withcontinuous paths (Rt)t≥0 (the rates can be negative, too) such that the followingconditions hold:

(1) Bt is a well-defined predictable, strictly positive process

Bt := exp(∫ t

0

Rsds)


Furthermore we assume that Bt

Bt+hare integrable with respect to Q for

t ≥ 0 and h ≥ 0.(2) The measure Q is locally equivalent to P , i.e. Q|Ft ∼ P |Ft for t ≥ 0.(3) The discounted processes B−1

t P (t, T ) are Q-martingales for 0 ≤ t ≤ T (inparticular elements of L1(Ω,F , Q)).

(4) For t ≥ 0 the curve T 7→ P (t, T ) for T ≥ t is C1, i.e. we can define thederivative

f(t, T ) := − ∂

∂TlnP (t, T )

for 0 ≤ t ≤ T as an adapted process with continuous paths and T 7→f(t, T ) continuous for each t ≥ 0 and T ≥ t.

The no arbitrage condition yields therefore

B−1t P (t, T ) = EQ(B−1

s P (s, T )|Ft)

for s ≥ t. Taking s = T we get by the given normalization and adaptedness of Bt

the representation

P (t, T ) = EQ(Bt

BT|Ft)

= EQ(exp(−∫ T

t

Rsds)|Ft)

of the price processes for 0 ≤ t ≤ T in the martingale measure Q.The forward rate is f(t, T ) satisfies the following formula

P (t, T ) = exp(−∫ T

t

f(t, s)ds)

We obtain under regularity assumptions (dominated convergence) the basic relationf(t, t) = rt for t ≥ 0. The yield process

Y (t, T ) :=1

T − t

∫ T

t

f(t, s)ds

is well defined for 0 ≤ t ≤ T .

3. Classical Interest Rate models

There are two classical interest rate models, which have been widely used. Weshall calculate directly in the martingale measure Q to obtain the relevant formulas.Certainly an interest rate model is given in the martingale measure by fixing theshort rate process in the martingale measure.

3.1. The Vasicek model. In this model the short rate process is given amean revering process (Rt)t≥0 with parameters β, ρ, b,

dRt = (b− βRs)ds+ ρdWt,

R0 ∈ R.

where• (Wt)t≥0 is a one-dimensional Brownian motion on the natural filtered

probability space.• ρ ≥ 0 is a volatility parameter.• β ≥ 0 is the mean reversion parameter.

3. CLASSICAL INTEREST RATE MODELS 11

• b denotes the level of mean reversion.

In this case we can explicitly write down the solution of the stochastic differ-ential equation by variation of constants

Rt = R0 exp(−βt) +∫ t

0

exp(−β(t− s))bds+∫ t

0

exp(−β(t− s))ρdWs.

We obtain a Gaussian process with expectation

E(Rt) = R0 exp(−βt) + b1− exp(−βt)

β

var(Rt) =∫ t

0

exp(−2β(t− s))ρ2ds

= ρ2 1− exp(−2βt)2β

.

Furthermore this process is Markovian process, which yields that

P (t, T ) = EQ(exp(−∫ T

t

Rsds)|Ft)

= G(Rt, T − t)

G(R0, T ) := EQ(exp(−∫ T

0

Rsds)).

By means of Laplace transforms of Gaussian random variables we are able to cal-culate the relevant quantities:

EQ(exp(−∫ T

0

Rsds)) = exp(−E(∫ T

0

Rsds) +12

var(∫ T

0

Rsds)).

We obtain by Ito’s Lemma and Fubini’s theorem

E(∫ T

0

Rsds) = R01− exp(−βT )

β+ b

T

β− b

1− exp(−βT )β2

var(∫ T

0

Rsds) = cov(∫ T

0

Rsds,

∫ T

0

Rudu)

=∫ T

0

∫ T

0

cov(Rs, Ru)dsdu

cov(Rs, Ru) = ρ2 exp(−β(s+ u))EQ(∫ s

0

exp(βr)dWs

∫ u

0

exp(−βr)dWr)

= ρ2 exp(−β(s+ u))exp(2β(s ∧ u))− 1

2β

var(∫ T

0

Rsds) =ρ2T

β2− ρ2

β3(1− exp(−βT ))− ρ2

2β3(1− exp(−βT ))2.


Hence we finally obtain the following formula for the logarithm of the prices

logP (t, T ) = −Rt1− exp(−β(T − t))

β− b(T − t)

β+ b

1− exp(−β(T − t))β2

+

ρ2(T − t)2β2

− ρ2

2β3(1− exp(−β(T − t)))

− ρ2

4β3(1− exp(−β(T − t)))2

= −R∞(T − t) +1β

((R∞ −Rt)(1− exp(−β(T − t)))−

ρ2

4β2(1− exp(−β(T − t)))2)

R∞ =b

β− ρ2

2β2

and finally

f(t, T ) = R∞ − (R∞ −Rt) exp(−β(T − t))+

+ρ2

4β2(1− exp(−β(T − t))) exp(−β(T − t))

= R∞ −R∞ exp(−β(T − t))+

+ρ2

4β2(1− exp(−β(T − t))) exp(−β(T − t)) +Rt exp(−β(T − t))

= A(t, T ) +RtB(t, T )

for the forward rates, 0 ≤ t ≤ T .

3.2. The Cox-Ingersoll-Ross model. In this model the short rate processis given by a square root process

dRt = (b− βRt)dt+ ρ√RtdWt,

R0 ≥ 0.

Even though the solution can not be described as direct as for the Vasicek model,we are able to prove existence of unique solutions with continuous paths. Again wecan calculate Laplace transforms easily - we consider a Feynman-Kac approach:

F (t, R0) := EQ(exp(−λRt − µ

∫ t

0

Rsds))

satisfies the equation

∂

∂tF (t, x) =

ρ2x

2∂2

∂x2F (t, x) + (b− βx)

∂

∂xF (t, x)− µxF (t, x)

F (0, x) = exp(−λx)

for t, x ≥ 0. If we are able to find C2-solutions with nice integrability properties,we are done. Therefore we apply due to the affine structure the following Ansatz

F (t, x) = exp(−bφ(t)− xψ(t)),

4. HJM APPROACH 13

which leads to Riccati equations for ψ and φ,

ρ2

2ψ(t)2 + βψ(t)− µ = −ψ′(t)

ψ(t) = φ′(t).

These equations have nice solutions for b, σ ≥ 0 and β > 0 (the relevant cases forfinance).

We are led to the following expressions for prices:

P (0, T ) = EQ(exp(−∫ T

0

Rsds))

= exp(−bφ(T )−R0ψ(T ))

with

φ(t) = − 2ρ2

log(2(√β2 + 2ρ2 exp( t(

√β2+2ρ2+β)

2 )√β2 + 2ρ2 − β + exp(t(

√β2 + 2ρ2))(

√β2 + 2ρ2 + β)

,

ψ(t) =2(exp(t(

√β2 + 2ρ2))− 1)√

β2 + 2ρ2 − β + exp(t(√β2 + 2ρ2))(

√β2 + 2ρ2 + β)

.

Hence we are led to the following presentation in the martingale measure by Marko-vianity

P (t, T ) = exp(−bφ(T − t)−Rtψ(T − t)),

which in turn leads to the following formula for forward rates

f(t, T ) = bψ(T − t) +Rtψ′(T − t).

Again we obtain a similar structure for the forward curves.

Definition 8. Let (Rt)t≥0 be a sufficiently regular short rate process (such thatforward rates exist) and assume that there are smooth functions A,B : R≥0×R≥0 →R such that

f(t, T ) = A(t, T ) +RtB(t, T ).

Then we say we are given a 2-dimensional realization of forward rates (time t andshort rate Rt as variables).

As we saw in the previous calculations, the Vasicek and the Cox-Ingersoll-Rossmodel admit 2-dimensional realizations.

4. HJM approach

We are given stochastic basis (Ω,F , Q) together with a d-dimensional Brow-nian motion (W 1

t , . . . ,Wdt )t≥0. We assume that there are predictable processes

(α(t, T ))0≤t≤T and (βi(t, T )0≤t≤T for T ≥ 0 such that

df(t, T ) = α(t, T )dt+d∑

i=1

βi(t, T )dW it

is a well-defined process for T ≥ 0, which is almost surely continuous in maturityT ≥ t. What are the conditions on α, β1, . . . , βd such that the associated discountedprice processes are Q-martingales.


Theorem 12. Under the above conditions the discounted price processes

(B−1t P (t, T ))0≤t≤T

are local Q-martingales if and only if

α(t, T ) =d∑

i=1

βi(t, T )∫ T

t

βi(t, s)ds.

Proof. We shall apply Ito’s formula for the process X(t, T ) :=∫ T

tf(t, s)ds

locally and obtain

dX(t, T ) = f(t, t)dt+ (∫ T

t

α(t, s)ds)dt+d∑

i=1

(∫ T

t

βi(t, s)ds)dW it

for the semimartingale decomposition. Hence the process (B−1t P (t, T ))0≤t≤T is a

local Q-martingale if and only if∫ T

t

α(t, s)ds =12

d∑i=1

(∫ T

t

βi(t, s)ds)2,

which leads by derivation with respect to T the desired equation.

Hence the equation for the forward rate process in the martingale measurereads

df(t, T ) =d∑

i=1

βi(t, T )∫ T

t

βi(t, s)dsdt+d∑

i=1

βi(t, T )dW it .

To obtain an equation which can be interpreted as stochastic differential equa-tion rather than as system of equations, we consider the fundamental Musielaparametrization:

T = t+ x

for 0 ≤ t ≤ T . We introduce time to maturity x instead of absolute maturities.This leads to the following equation for r(t, x) := f(t, x+ t),

dr(t, x) =∂

∂xr(t, x) +

d∑i=1

βi(t, x+ t)∫ x

0

βi(t, y + t)dydt+d∑

i=1

βi(t, x+ t)dW it .

The advantage is that we are able to interpret this equation on a Hilbert space offorward rate curves.

5. The HJM equation as SDE with values in a Hilbert space

We are now able to interpret the equation for the time evolution of forwardrates as stochastic differential equation with values in a Hilbert space H. We haveto find a Hilbert space of forward rate curves for this purpose such that the followingconditions hold:

• H is a separable Hilbert space of continuous functions and point evalua-tions are continuous with respect to the topology of a Hilbert space.

• The shift semigroup (Ttr)(x) = r(t+x) is a strongly continuous semigroupon H with generator d

dx .

5. THE HJM EQUATION AS SDE WITH VALUES IN A HILBERT SPACE 15

• The map h 7→ S(h) with S(h)(x) := h(x)∫ x

0h(y)dy satisfies

||S(h)|| ≤ K||h||2

for all h ∈ H with S(h) ∈ H.We can prove the existence of such a Hilbert space by the following construction

(where we have concrete financial meanings of the different steps, too):Let w : R≥0 → [1,∞[ be a non-decreasing C1-function with

1w

13∈ L1(R≥0),

then we define

||h||w := |h(0)|+∫

R≥0

|h′(x)|w(x)dx

for all h ∈ L1locwith h′ ∈ L1

loc (where h′ denotes the weak derivative). We defineHw to be the space of all functions h ∈ L1

loc with h′ ∈ L1loc such that ||h||w < ∞.

We shall prove the relevant properties by four inequalities:(1) By Holder’s inequality we obtain

||h′||L1 =∫

R≥0

|h′(x)|w 12 (x)w−

12 (x)dx

≤ ||h||w||w−12 ||L1 = C1||h||w.

By means of the decomposition

h(x)− h(y) =∫ y

x

h′(z)dz,

we obtain||h||L∞ ≤ C2||h||w

and the fact that h(∞) is well defined.(2) Writing

Π(x) :=∫ ∞

x

w−1(y)dy

we obtain by monotonicity of w that

Π(x) ≤ w−23 (x)

∫ ∞

x

w−13 (y)dy ≤ w−

23 (x)||w− 1

3 ||L1

for x ≥ 0. Henceforward we conclude

||Π 32w||L∞ ≤ ||w− 1

3 ||L1 <∞

||Π 12 ||L1 ≤ ||w− 1

3 ||L1 ,

which finally leads to∫R≥0

|h(x)− h(∞)|dx =∫

R≥0

|∫ ∞

x

h′(y)w12 (y)w−

12 (y)dy|dx

≤∫

R≥0

(∫ ∞

x

|h′(y)|2w(y)dy)12 Π

12 (x)dx

≤ ||h||w||Π12 ||L1 ,

||h− h(∞)||L1 ≤ ||h||w||Π12 ||L1 = C3||h||w.


(3) The last inequality we shall need comes from the following similar consid-eration:∫

R≥0

|h(x)− h(∞)|4w(x)dx =∫

R≥0

|∫ ∞

x

h′(y)w12 (y)w−

12 (y)dy|4w(x)dx

≤∫

R≥0

(∫ ∞

x

|h′(y)|2w(y)dy)2Π2(x)w(x)dx

≤ ||h||4w||Π32w||L∞ ||Π

12 ||L1 ,

so

||(h− h(∞))4w||L1 ≤ ||h||4w||Π32w||L∞ ||Π

12 ||L1 = C4||h||w.

Therefore we are able to calculate all the necessary properties directly by ap-plying these inequalities. First we treat the shift semigroup

Ttr(x) = r(t+ x).

Take h ∈ Hwand a test function φ on R>0, then∫R≥0

Ttr(x)φ′(x)dx = −∫

R≥0

r′(x+ t)φ(x)dx

by partial integration. Hence (Ttr)′ = Ttr′, in particular it exists. Furthermore

||Ttr||2w = |h(t)|2 +∫

R≥0

|r′(x+ t)|2w(x)dx ≤ (C22 + 1)||h||w

by monotonicity of w. So T is a bounded semigroup. For g ∈ Hw with g′ ∈C∞0 (R≥0) we obtain the simple fact that

||Ttg − g||2w ≤ |g(t)− g(0)|2 + t2∫ 1

0

∫R≥0

|g′′(x+ st)|2w(x)dxds

≤ |g(t)− g(0)|2 + t2∫ 1

0

||Tstg′||wds→ 0

for t→ 0 by boundedness. A standard argument then leads to strong continuity.Next we show the third property on the bilinear map S: First we calculate

(Sh)′(x) = h′(x)∫ x

0

h(y)dy + h(x)2

for x ≥ 0, which leads us to the assertion that

Sh ∈ Hw if and only if h ∈ Hw and h(∞) = 0

for h ∈ Hw. If h(∞) 6= 0 the assertion is clear, for the other direction we shallapply

||Sh||2w =∫

R≥0

|h′(x)∫ x

0

h(y)dy + h(x)2|2w(x)dx

≤ 2∫

R≥0

|h′(x)∫ x

0

h(y)dy|2 + |h(x)|4w(x)dx

≤ 2(||h||L1 ||h||w + C4||h||4w) ≤ 2(C23 + C4)||h||4w,

which is the desired (uniform) assertion.

5. THE HJM EQUATION AS SDE WITH VALUES IN A HILBERT SPACE 17

We can also easily calculate the domain of the generatorA of the shift semigroupT , namely

D(A) = h ∈ Hw such that h′ ∈ HwAh = h′.

By the core theorem we know that the set of g ∈ Hw with g ∈ C∞0 (R≥0) is invariantunder translation and therefore a core of T . Furthermore this set is dense in Hw

and Ag = g′ by uniform convergence. Consequently the closure of the derivativeoperator coincides with A. The closure of d

dx in turn can be easily calculated bytaking gn → h with g′n → h in Hw, leads to h′ = h by

||h− h′||L∞ ≤ C2||h′ − h||w = 0.

Given a Hilbert space H with the above properties, such as Hw, we can formu-late a HJM-equation in the martingale measure by specifying the volatility structureσ1, . . . , σd : U → H0 locally Lipschitz,

drt = (d

dxrt +

d∑i=1

S(σi(rt)))dt+d∑

i=1

σi(rt)dW it(1.2)

r0 ∈ U.The following theory can be applied for the HJM-equation, but in fact for generalstochastic differential equations on Hilbert spaces.

CHAPTER 2

Finite dimensional Realizations

1. Locally invariant submanifolds

The definitions for sub-manifolds with boundary of Hilbert spaces carry overfrom the finite dimensional setting without any change (see [24] or [18] for all nec-essary details). We therefore plunge ourselves directly into the theory of locallyinvariant submanifolds and aim to prove the first fundamental result in this direc-tion. First we shall fix our setting. We are considering a separable Hilbert spaceand stochastic differential equations of the type

drt = (Art + α(rt))dt+d∑

i=1

σi(rt)dW it(2.1)

r0 = r,

where α, σ1, . . . , σd : U ⊂ H → H are smooth vector fields and A is the generatorof a strongly continuous semigroup S. In this case we have local existence of weakand mild solutions up to some strictly positive stopping time τ r.

Definition 9. Let M ⊂ H be a finite dimensional submanifold with boundary∂M . Then M is called locally invariant with respect to equation 2.1 if for all r ∈Mthere is a strictly positive stopping time ηr such that rt ∈M for 0 ≤ t ≤ τ r ∧ ηr.

Theorem 13. Let M ⊂ H be a finite dimensional submanifold with boundary∂M . Then M is called locally invariant with respect to equation 2.1 if and only ifM ⊂ D(A) and

µ(r) := Ar + α(r)− 12

d∑i=1

Dσi(r) · σi(r) ∈ TrM

σi(r) ∈ TrM

for all r ∈M \ ∂M and

µ(r) := Ar + α(r) +12

d∑i=1

Dσi(r) · σi(r) ∈ (TrM)≥0

σi(r) ∈ (TrM)0 = Tr∂M

for all r ∈ ∂M . Additionally the restrictions of the vector fields µ and σ1, . . . , σd

to M are smooth in the differentiable structure of M .

Remark 9. For the proof we shall need a special chart map, which existenceis easily established by the following construction. Take a submanifold M of finitedimension with boundary in a separable Hilbert space and fix r0 ∈ H: furthermore

19

20 2. FINITE DIMENSIONAL REALIZATIONS

we shall fix a subspace V ⊂ H of dimM , such that the orthogonal projection prVon V restricts to an isomorphism on Tr0M , i.e.

prV |Tr0M : Tr0M → V

is an isomorphism. Fixing a half-space V≥0 on V we can furthermore guarantee thatinward-pointing elements are mapped to V≥0. We take now any sub-manifold chartφ1 around r0 ∈M with φ1(r0) = 0, then prV (φ−1

1 |W (pr1(0)×0 − r0) : W → V islocally invertible by the inverse function theorem due to the previous assumption.We denote the inverse by ψ1. Hence we can construct the following chart map

φ(r0 + v) = (prV (v), prV ⊥(r0 + v − φ−11 (ψ1(prV (r))× 0))

locally. This map is invertible by the very construction. Furthermore we have

φ(r0 + v) = (prV (v), 0)

for r0 + v ∈M locally, since r0 + v ∈M means that there is w ∈W with r0 + v =φ−1

1 (w, 0), and w corresponds precisely to prV (v) via the above construction (sor0 + v − φ−1

1 (w, 0) = 0).

Proof. We denote n := dimM and fix r0 ∈ M . We assume first that thediffusion lets the sub-manifold M locally invariant. We can find an orthonormalsystem ξ1, . . . , ξn ∈ D(A′) (which generates the subspace V , such that inward-pointing maps to inward-pointing) and a submanifold chart φ : U1 → H by theprevious remark, where U1 is an open neighborhood of r0, φ(M ∩ V ) = W × 0with W open in V≥0, such that

φ(r0 + v) = pr〈ξ1,...,ξn〉(v)

for i = 1, . . . , n for r0 + v ∈ U1 ∩ M . Here pr〈ξ1,...,ξn〉 denotes the orthogonalprojection on the space V generated by ξ1, . . . , ξn. In words: we restrict the affineprojection on 〈ξ1, . . . , ξn〉 to M , which locally yields a diffeomorphisms and whichcan be embedded in a submanifold chard φ. Since (rt)0≤t≤τr0 is a weak solution ofequation (2.1), we obtain semimartingales (〈rt, ξi〉)0≤t≤τr0 for i = 1, . . . , n,

〈rt, ξi〉 = 〈r0, ξi〉+∫ t

0

〈rs, A′ξi〉 ds+∫ t

0

〈α(rs), ξi〉 ds+d∑

j=1

∫ t

0

⟨σj(rs), ξi

⟩dW j

s

for i = 1, . . . , n up to some stopping time τ r0 (notice that ξi ∈ D(A′). Hence – bylocal invariance, after all rt ∈M almost surely for t ≤ τ r0 –

φ(rt) =∫ t

0

⟨φ−1 φ(rs), A′ξi

⟩ds+

∫ t

0

⟨α(φ−1 φ(rs)), ξi

⟩ds+

+d∑

j=1

∫ t

0

⟨σj(φ−1 φ(rs)), ξi

⟩dW j

s ,

and therefore we can define smooth vector fields a, b1, . . . , bd on W

ai(x) :=⟨φ−1(x, 0), A′ξi

⟩+⟨α(φ−1(x, 0)), ξi

⟩bji (x) :=

⟨σj(φ−1(x, 0)), ξi

⟩for i = 1, . . . , n. Notice that the vector-fields have to inward-pointing, respectivelyparallel to the boundary (at the boundary) by construction. Now we can apply

1. LOCALLY INVARIANT SUBMANIFOLDS 21

Ito’s formula for ψ := φ−1|W×0.

rt = r0 +∫ t

0

Dψ(φ(rs)) · a(φ(rs))ds+

+12

d∑j=1

∫ t

0

D2ψ(φ(rs)) · (bj(φ(rs)), bj(φ(rs)))ds+

+d∑

j=1

∫ t

0

Dψ(φ(rs)) · bj(φ(rs))dW js

up to some strictly positive stopping time τ . Hence by uniqueness of the semi-martingale decomposition

〈rs, A′ξ〉+ 〈α(φ(rs)), ξ〉 = 〈Dψ(φ(rs)) · a(φ(rs))+

12D2ψ(φ(rs)) · (bj(φ(rs)), bj(φ(rs))), ξ

⟩⟨σj(rs), ξ

⟩=⟨Dψ(φ(rs)) · bj(φ(rs)), ξ

⟩for all ξ ∈ D(A′) and 0 ≤ s ≤ τ ∧ τ r almost surely. By continuity of pathswe can make this assertion valid outside a null set. Therefore we conclude thatξ 7→ 〈r0, A′ξ〉 is continuous with respect to ξ, since the right hand side is continuouswith respect to ξ. Whence r0 ∈ D(A) and

Ar0 + α(r0)−12D2ψ(φ(r0)) · (bj(φ(r0)), bj(φ(r0))) = Dψ(φ(r0)) · a(φ(r0)) ∈ Tr0M,

σj(r0) = Dψ(φ(r0)) · bj(φ(r0)) ∈ Tr0M

By the second equation we obtain the correction term, since for a smooth curve cwith values in M , c(0) = r0and c′(0) = bj(r0) we obtain

D2ψ(φ(r0)) · (bj(φ(r0)), bj(φ(r0))) =d

ds|s=0Dψ(φ(c(s))) · bj(c(s))−

−Dψ(φ(r0)) ·d

ds|s=0b

j(c(s))

=d

ds|s=0σ

j(c(s))−Dψ(φ(r0)) ·d

ds|s=0b

j(c(s))

= Dσj(r∗) · σj(r∗)−Dψ(φ(r∗)) · dds|s=0b

j(c(s)).

This yields what we expectµ(r0) ∈ Tr0M.

The conditions on the directions at the boundary are true by construction. Sincer0 ∈ M was chosen arbitrarily, we have proved one direction. The smoothnessof the vector fields is also clear from the above defining equations. Taking thetangent conditions on a manifold with boundary we can equally trace back the wayby defining a process on W with the vector fields a, b1, . . . , bd, since they are wellsmooth (with respect to x). Since M ⊂ D(A) we obtain a strong solution of theabove equation by Ito’s formula. The strong solution is a weak solution of equation2.1. By uniqueness this solution coincides with the given one and therefore weobtain the result.


Example 3. A popular forward curve-fitting method is the Svensson [32] family

GS(x, z) = z1 + z2e−z5x + z3xe

−z5x + z4xe−z6x.

It is shown in [14] that the only non-trivial interest rate model that is consistentwith the Svensson family is of the form

(2.2) rt = Z1t g1 + · · ·+ Z4

t g4,

where

g1(x) ≡ 1, g2(x) = e−αx, g3(x) = xe−αx, g4(x) = xe−2αx,

for some fixed α > 0. Moreover,

Z1t ≡ Z1

0 , Z3t = Z3

0e−αt, Z4

t = Z40e−2αt (Z4

0 ≥ 0)

and Z2 satisfies

(2.3) dZ2t =

(Z3

t + Z4t − αZ2

t

)dt+

√αZ4

t dWt.

We shall now describe a HJM-equation which leaves the linear submanifold withboundary

M := 4∑

j=1

zjgj with zi ∈ R and z4 ≥ 0

locally invariant. We take a Hilbert space H of forward curves, a candidate for σis given, on U := ` > 0, by

σ(h) =√α`(h)g2,

where ` is some continuous linear functional on H with `(g1) = `(g2) = `(g3) = 0and `(g4) = 1. Straightforward calculations show, for h ∈ U ,

µ(h) =d

dxh+ `(h)g2 − `(h)g2

2 .

(the clue is that ` σ ≡ 0). Hence for h =∑4

j=1 zjgj, we obtain

µ(h) = −αz2g2 + z3g2 − αz3g3 + z4g22 − 2αz4g4 + z4g2 − z4g

22

= (−αz2 + z3)g2 − αz3g3 + (−2αz4)g4.

Consequently µ and σ satisfy the tangent conditions at all points of M .

2. Montgommery-Zippin Results on Semiflows

In order to conclude that the (minimal) locally invariant sub-manifolds lie infact inD(A∞) := ∩n≥0D(An) and that there differentiable structure is in fact differ-entiable with respect to the natural topology on D(A∞) ⊂ H, we shall need severalconsiderations on local (semi-)flows of maps in finite dimensional sub-manifolds ofHilbert spaces.

We shall apply the fundamental theorem on local invariance and several resultsgeneralizing the work of Dean Montgommery and Leon Zippin in the 1950s. Letk ≥ 1 be given. We consider a Banach space X and a continuous local semiflow Flof Ck-maps on an open subset V ⊂ X , i.e.

(1) There is ε > 0 and V ⊂ X open with Fl : [0, ε[×V → X a continuousmap.

(2) Fl(0, x) = x and Fl(s, F l(t, x)) = Fl(s + t, x) for s, t, s + t ∈ [0, ε[ andx, F l(t, x) ∈ V .

2. MONTGOMMERY-ZIPPIN RESULTS ON SEMIFLOWS 23

(3) The map Flt : V → X is Ck for t ∈ [0, ε[.

To shorten terminology we say that “Fl” is a continuous local semiflow of Ck-maps on X if for any x ∈ X there is an open neighborhood V ⊂ X of x and acontinuous local semiflow Fl = Fl(V ) of Ck-maps on V , such that Fl(V1) = Fl(V2) onV1 ∩ V2. Continuous local semiflows of Ck-maps appear naturally as mild solutionsof nonlinear evolution equations (see Appendix A). The continuous local semiflowFl is called Ck or local Ck-semiflow if Fl : [0, ε[×V → X is Ck.

We assume that we are given a finite-dimensional Ck-submanifold M withboundary of X such that M is locally invariant for Fl, i.e. for every x ∈ M ∩ Vthere is δx ∈]0, ε[ such that Fl(t, x) ∈M for 0 ≤ t ≤ δx. In this case Fl restricts ina small open neighborhood of any point x ∈M ∩ V to a continuous local semiflowof Ck-maps on M , where we make the restriction precise. A continuous localsemiflow of Ck-maps on M is defined as above, the same for local Ck-semiflows:notice however that the manifold might have a boundary. By TxM we denote the(full) tangent space at x ∈ M , even at the boundary. By (TxM)≥0 we denote thehalfspace of inward pointing tangent vectors for x ∈ ∂M . The boundary subspaceof this halfspace is the tangent space of the Ck-submanifold (without boundary)∂M , these are the tangent vectors parallel to the boundary.

We shall prove that the restriction of a continuous local semiflow of Ck-mapsFl to a Ck-submanifold with boundary M is jointly Ck and can in particularbe embedded in a local Ck-flow around any interior point of M . We shall applyclassical methods developed to solve the fifth Hilbert problem. Nevertheless we haveto face the difficulty that Fl is only a continuous local semiflow. We can prove theresult under a weak assumption, which will always be satisfied with respect to ourapplications.

We first cite the classical results from Dean Montgomery and Leo Zippin anddraw a simple conclusion, which illustrates, what are going to do, namely provinga non-linear version of the above example.

Theorem 14. Let M be a finite-dimensional Ck-manifold and Fl : R×M →Ma continuous flow of Ck-maps on M , then Fl is a Ck-flow on M .

Example 4. Let S be a strongly continuous group on a Banach space X andassume that M is a locally S-invariant finite-dimensional Ck-submanifold of X.Then M ⊂ D(Ak), where A denotes the infinitesimal generator of S, and therestriction of A to M is a Ck−1-vector field on M .

We end this section by the announced lemma. Let M be a finite-dimensionalCk-submanifold with boundary and let M be locally invariant for Fl, as definedabove. We denote by Rn

≥0 the halfspace x ∈ Rn; xn ≥ 0, consequently R≥0 is thepositive halfline including 0.

Lemma 2. For every x ∈M ∩ V there exists an open neighborhood V ′ ⊂ X ofx and ε′ > 0, such that Fl(t, y) ∈M for all (t, y) ∈ [0, ε′[×(V ′ ∩M).

Proof. Take x ∈ M and a Ck-submanifold chart u : U ⊂ X → X withu(U ∩M) = 0×W ⊂ 0×Rn

≥0, where U ⊂ U ⊂ V is open and x ∈ U , W ⊂ Rn≥0

is open, convex. Here n denotes the dimension of M . We may assume that u has acontinuous extension on U with u(U ∩M) = u(U ∩M) = 0 ×W by restrictionof U . The closure of U ∩M is taken in M .


For y ∈ U define the lifetime in U ∩MT (y) := sup0 < t < ε | ∀0 ≤ s < t : Fl(s, y) ∈ U ∩M, .

By continuity of Fl we have Fl(T (y), y) ∈ U ∩M \ (U ∩ M) if T (y) < ε. Weclaim that there exists an open neighborhood V ′ ⊂ U of x in X and ε′ > 0 suchthat T (y) ≥ ε′ for all y ∈ V ′. Indeed, otherwise we could find a sequence (xn) inU ∩M with xn → x and ε > T (xn) → 0. But this means that u(Fl(T (xn), xn)) ∈0×(W \W ) converges to u(Fl(0, x)) = u(x) ∈ 0×W , a contradiction. Whencethe claim, and the lemma follows.

Since we are treating local questions as differentiability, we can – without anyrestriction – assume that f : [0, ε[×V → Rn

≥0 is a given continuous local semiflowof Ck-maps, where V is open, convex in Rn

≥0. We do not make a difference innotation between right derivatives and derivatives, even though on the boundarypoints in space or time, respectively, we only calculate right derivatives. We shallalways assume in this section that f is continuous and f(t, .) is Ck for all t ∈ [0, ε[,for some k ≥ 1. We shall write f(t, x) = (f1(t, x), ..., fn(t, x)) for (t, x) ∈ [0, ε[×V .

Condition 1 (crucial). We assume that for any x ∈ V there is εx > 0 suchthat Dxf(t, x) is invertible for 0 ≤ t ≤ εx (Dxf denotes the derivative with respectto x).

Lemma 3. The mapping (t, x) 7→ Dxf(t, x) is continuous.

Proof. For the proof we proceed from the Baire category Theorem. We thenhave the following result:

Let Z be any compact interval, V the open set in Rn≥0 and let F : Z × V → R

be a continuous real valued function, such that F (g, .) is C1 for any g ∈ Z. Givena ∈ V and 1 ≤ i ≤ n, the set of points g0 ∈ Z such that ∂

∂xiF is continuous at

(g0, a) is dense in Z, even more, the set where it is not continuous is of first categoryin Z.

Let now a ∈ V be fixed, then the set of points t0 ∈ [0, ε[ such that fij := ∂∂xi

fj

is continuous at (t0, a), for all 1 ≤ i, j ≤ n, is everywhere dense in [0, ε[. We shalldenote this set by Ia. In addition the determinant det(fij) is continuous at thesepoints, too. We want to show now that for fixed a ∈ V the mappings fij arecontinuous at (0, a). Notice that the determinant at any point of continuity (t0, a),with t0 ∈ Ia small enough, is bounded away from zero in a neighborhood.

We fix a ∈ V , then for t0 ∈ [0, ε[

f(t0 + h, a+ y) = f(t0, f1(h, a+ y), ..., fn(h, a+ y))

for h ≥ 0 and y ∈ Rn≥0, both sufficiently small, hence

Dxf(t0 + h, a+ y) = Dxf(t0, f(h, a+ y)) ·Dxf(h, a+ y).

There is t0 ∈ Ia such that Dxf(t0, z) is invertible in a neighborhood of a, hence

Dxf(t0, f(h, a+ y))−1 ·Dxf(t0 + h, a+ y) = Dxf(h, a+ y)

and therefore

id = limh↓0,y→0

Dxf(t0, f(h, a+ y))−1 ·Dxf(t0 + h, a+ y) = limh↓0,y→0

Dxf(h, a+ y)

by continuity of Dxf at (t0, a), continuity of f in both variables and the continuityof the inversion of matrices. So 0 ∈ Ia for all a ∈ V .

2. MONTGOMMERY-ZIPPIN RESULTS ON SEMIFLOWS 25

Now we can conclude for arbitrary t ∈]0, ε[ in the following way:

Dxf(t+ h, a+ y) = Dxf(t, f(h, a+ y)) ·Dxf(h, a+ y)

for h ≥ 0 and y ∈ Rn≥0 sufficiently small, hence by continuity at (0, a)

limh↓0,y→0

Dxf(t+ h, a+ y) = limh↓0,y→0

Dxf(t, f(h, a+ y)) ·Dxf(h, a+ y) = Dxf(t, a).

For left continuity we apply

Dxf(t, a+ y) = Dxf(h, f(t− h, a+ y)) ·Dxf(t− h, a+ y)

for h ≥ 0 and y ∈ Rn≥0 sufficiently small, hence by continuity of Dxf at (0, a) and

(0, f(t, a)), the continuity of Dxf in the second variable and the existence of theinverse for small h

limh↓0,y→0

Dxf(t−h, a+y) = limh↓0,y→0

Dxf(h, f(t−h, a+y))−1·Dxf(t, a+y) = Dxf(t, a).

Consequently the desired assertion holds.

In the next step we shall show that there is a derivative at 0.

Lemma 4. The right-hand derivative ddtf(t, x)|t=0 exists for x ∈ V , and for

small h ≥ 0 we have the formula

f(h, x)− x =∫ h

0

Dxf(t, x)dt · ( ddtf(0, x)).

Moreover, ddtf(t, .)|t=0 : V → Rn is continuous.

Proof. We may differentiate with respect to x under the integral sign by thelemma and uniform convergence, so

T (h, x) :=∫ h

0

f(t, x)dt

DxT (h, x) :=∫ h

0

Dxf(t, x)dt.

By the mean value theorem we obtain

T (h, y)− T (h, x) = DxT (h, x)(y − x),

where x ∈ [x, y]. Now we take y = f(p, x), then

T (h, y)− T (h, x) =∫ h+p

p

f(t, x)dt−∫ h

0

f(t, x)dt

=∫ h+p

h

f(t, x)dt−∫ p

0

f(t, x)dt,

which finally yields1p(∫ p

0

f(t+ h, x)dt−∫ p

0

f(t, x)dt) = DxT (h, x)[1p(f(p, x)− x)].

This equation can be solved by joint continuity of (h, z) 7→ 1h

∫ h

0Dxf(t, z)dt: we

obtain for small h and a compact set in x that the expression is in a small neigh-borhood of the identity matrix. So inversion leads to the desired result and thento the given formula.

The formula asserts again by inversion, that the derivative is continuous withrespect to x.


By the semigroup-property and the chain rule, the result of Lemma can beextended for 0 < t < ε, and the derivative of f(·, x) exists for all t ∈ [0, ε[. Indeed,

limp↓0

f(t+ p, x)− f(t, x)p

= limp↓0

f(p, f(t, x))− f(t, x)p

=d

dtf(0, f(t, x))

limp↓0

f(t, x)− f(t− p, x)p

= limp↓0

f(p, f(t− p, x))− f(t− p, x)p

= limp↓0

1p(∫ p

0

Dxf(t, f(t− p, x))dt)d

dtf(0, f(t− p, x))

=d

dtf(0, f(t, x))

by the previous lemmas. Consequently for small h ≥ 0

(2.4)d

dtf(t, x) =

d

dtf(0, f(t, x)) = (

∫ h

0

Dxf(s, f(t, x))ds)−1 · (f(h, x)− x).

In particular (t, x) 7→ ddtf(t, x) is continuous in both variables on the whole domain

of definition.

Lemma 5. The semiflow f is Ck in both variables.

Proof. If f(t, .) is Ck for t ∈ [0, ε[, then the r-jet

(f(t, x0), Dxf(t, x0) · x1, ..., Drxf(t, x0) · x1 · ... · xr)

for (t, x0, x1, ..., xr) ∈ [0, ε[×V ×Rn× ...×Rn is a local semiflow of Ck−r-maps, for0 ≤ r ≤ k − 1. For r = 1, the 1-jet is a continuous, local semiflow of Ck−1-maps.Assume that for r < k the r-jet is a continuous, local semiflow then, again, the(r + 1)-jet is continuous. By induction

(t, x) 7→ Drxf(t, x)

is continuous in both variables for 0 ≤ r ≤ k.If we apply the above results to the r-jet for r < k, we conclude by equation

(2.4) that Drxf(t, x) can be (k−r) times differentiated with respect to the t-variable,

and these derivatives are continuous. Hence f is Ck in both variables.

Theorem 15. Let k ≥ 1 be given and let Fl : [0, ε[×U →M be a local semiflowon a finite-dimensional Ck-manifold M with boundary, which satisfies the followingconditions:

(1) The semiflow Fl : [0, ε[×U →M is continuous with U ⊂M open.(2) The mapping Fl(t, .) is Ck.(3) For fixed x ∈ U there exists εx > 0 such that TxFl(t, .) is invertible for

0 ≤ t ≤ εx.Then Fl is Ck and for any x0 ∈ U\∂M there is a local Ck-flow F l :]−δ, δ[×V →

M with V ⊂ U \ ∂M open around x0 and δ ≤ ε such that Fl(y, t) = F l(y, t) fory ∈ V and 0 ≤ t ≤ δ. This also holds for the smooth case (k = ∞).

Proof. By the previous lemmas the map Fl : [0, ε[×U →M is a Ck-semiflowon M . We fix x0 ∈ U \ ∂M , then there is 0 < δ < ε and W ⊂ U open inM \ ∂M , such that (t, x) 7→ (t, F l(t, x)) is Ck-invertible on [0, δ[×W by the Ck-inverse function theorem on manifolds with boundary. We then choose an openneighborhood V ⊂ ∩0≤t<δFl(t,W ) of x0 in M \ ∂M . Therefore we can defineF l(−t, y) := Fl(., .)−1(t, y) for t ∈ [0, δ[ and y ∈ V . Since this is the unique

3. M ⊂ D(A∞) AS SUB-MANIFOLD 27

solution in z of the Ck-equation Fl(t, z) = y, we obtain a Ck-map F l. The flowproperty holds by uniqueness, too. Notice that V can be chosen independent ofk.

Remark 10. Remark that for evolutions (which correspond in the differentiablecase to time-dependent vector fields) we can pass to the extended phase space andapply the results thereon.

3. M ⊂ D(A∞) as sub-manifold

We consider now an application of the Montgommery-Zippin results from theprevious section:

Theorem 16. Let M ⊂ U ⊂ H be a smooth submanifold with boundary ofdimension dimM = n, and assume that there are pointwise linearly independentBanach map vector fields X1, . . . , Xn−1 : H → D(A∞) (see section 3 for the precisedefinitions) on U , which are tangent to M and parallel to the boundary at boundarypoints. Furthermore we are given a generator A of a strongly continuous semi-groupS and α an additional Banach map vector field on U . We define

Xn(x) := Ax+ α(x)

for x ∈ D(A), and assume M ⊂ D(A) and µ(x) ∈ TxM for x ∈ M and µ(x) ∈(TxM)≥0 for x ∈ ∂M (these are exactly the conditions of Theorem 13). Then weobtain that M ⊂ D(A∞) and that M is a sub-manifold with boundary of D(A∞).

Proof. By Theorem 21 and 22 we obtain local, smooth flows FlXi , for i =1, . . . , n, associated to the vector fields X1, . . . , Xn−1 and a local, continuous semi-flow associated to Xn, which restrict to D(A∞) to local, smooth (semi-)flows. ByTheorem 13 we know that the flows restrict locally (up to some small time) tothe manifold M , where they constitute local, continuous flows of C∞-maps (forthe semi-flow this can easily be seen by investigating the first variation equation).By Theorem 15 we can conclude that those flows are smooth in the differentiablestructure of M , also with respect to time t. By [17], Th. 3.1, we can then concludethat M ⊂ D(A∞) and

(u1, . . . , un) 7→ FlX1t1 · · · FlXn

tn(x0)

for x0 ∈ M \ ∂M constitutes a local parametrization for M (it is obviously bijec-tive on M locally). By smoothness of this map to D(A∞), we conclude that Madmits smooth submanifold charts in D(A∞) (see section 3 for the construction ofa submanifold chart from a parametrization).

CHAPTER 3

Frobenius Theory

1. Geometric and Analytic Methods

We are treating the problem of existence of finite dimensional realizations forequations of the type (1.2). In Section 2, Th.16, we explained that it is sufficient tosolve the deterministic consistency problem on the Frechet space D(A∞). Thereforewe shall work on D(A∞) in the sequel.

First we shall provide the relevant conditions on the volatility structure, whichwill be explained in the sequel. We work on a separable Hilbert space H, where astrongly continuous semi-group S with generator A is acting upon.

(A1): We have

σi(r) = φi(`(r)), 1 ≤ i ≤ d,

where ` ∈ L(H,Rp), for some p ∈ N, and φ1, . . . , φd : Rp → D(A∞) aresmooth and pointwise linearly independent maps. Hence

σj : H → D(A∞0 )

are Banach maps. Furthermore we are given φ0 : Rp → D(A∞) and define

µ(r) := Ar + φ0(r)−12

d∑i=1

dσi(r) · σi(r)

= Ar + α(r)

for r ∈ D(A).(A2): For every q ≥ 0, the map

(`, ` (d/dx), . . . , ` (d/dx)q) : D ((d/dx)∞) → Rp(q+1)

is open.(A3): A is unbounded; that is, D(A) is a strict subset of H. Equivalently,A : D(A∞) → D(A∞) is not a Banach map.

We shall usually think of A = ddx , but the ideas are completely general and can

be formulated for the general case.First we give a guided tour through the proof:

Analysis: Analysis on Frechet spaces is a subtle subject since – given amap f : E → F on Frechet spaces with derivative df : U → L(E,F )– it is not clear how to define the second derivative consistently due tothe fact that L(E,F ) is no more a Frechet space in general. Thereforesome new concepts enter the scenery, which are even for classical analysisa considerable simplification.

29

30 3. FROBENIUS THEORY

Geometry: Given vector fields µ, σ1, . . . , σd on an open subset U with as-sociated flows Fl, the map

(u0, . . . , ud) 7→ Flµu0 · · · Flσ

d

ud(r∗)

is the obvious candidate for a parametrization of a submanifold withboundary at r∗ ∈ U tangent to µ, σ1, . . . , σd, if we expect the dimensionto be d + 1. To formulate the algebraic obstructions for this assertion,namely that the Lie brackets of the involved vector fields lie in the spanof the vector fields, e.g.

[µ, σ](r) = λ0(r)α(r) +d∑

i=0

λi(r)σi(r)

for some smooth, real valued functions λi, we need an applicable analysisat hand.

Synthesis: To be able to apply the geometric results reasonably to our prob-lem we have to reinvestigate the ingredients of equation (1.2), namely theclass of involved vector fields, to obtain finally a fairly general classificationresult. We shall see that the vector fields µ and σj have different analyticproperties and are rarely linearly dependent under conditions (A1)–(A3).

2. Analysis

For the purposes of analysis on open subsets of Frechet spaces we shall followtwo equivalent approaches. The classical Gateaux-approach as outlined in [20] andso called “convenient analysis” as in [24]. On Frechet spaces these two notions ofsmoothness coincide and convenient calculus is the appropriate extension of analysisto more general locally convex spaces. The combinations of these methods allowsimple and elegant calculations. The main advantage of convenient calculus ishowever, that one can give a precise analytic meaning (in simple terms) to geometricobjects on Frechet spaces such as vector fields or differential forms (see [24]), whichdo not lie in Frechet spaces generically. First we recall the definitions of Gateaux-Cn-calculus.

Definition 10. Let E,F be Frechet spaces and U ⊂ E an open subset. A mapP : U → F is called Gateaux-C1 if

dP (f)h := limt→0

P (f + th)− P (f)t

exists for all f ∈ U and h ∈ E and dP : U × E → F is a continuous map.

For the definition of Gateaux-C2-maps the ambiguities of calculus on Frechetspaces already appear. Since there is no Frechet space topology on the vector spaceof continuous linear mappings L(E,F ) one has to work by point evaluations:

Definition 11. Let E,F be Frechet spaces and U ⊂ E an open subset. A mapP : U → F is called Gateaux-C2 if

d2P (f)(h1, h2) := limt→0

DP (f + th2)h1 −DP (f)h1

t

exists for all f ∈ U and h1, h2 ∈ E and d2P : U ×E×E → F is a continuous map.Higher derivatives are defined in a similar way. A map is called Gateaux-smoothor Gateaux-C∞ if it is Gateaux-Cn for all n ≥ 0.

2. ANALYSIS 31

For the construction of differential calculus on locally convex spaces we needthe concept of smooth curves into locally convex spaces and the concept of smoothmaps on open subsets of locally convex spaces. We remark that already on Frechetspaces the situation concerning analysis was complicated and unclear until conve-nient calculus was invented (see [24], pp. 73–77, for extensive historical remarks).The reason for inconsistencies can be found in the fundamental difference betweenbounded and open subsets on locally convex vector spaces.

We denote the set of continuous linear functionals on a locally convex space Eby E′c. A subset B ⊂ E is called bounded if l(B) is a bounded subset of R for alll ∈ E′c. A multilinear map m : E1 × ...×En → F is called bounded if bounded setsB1× ...×Bn are mapped onto bounded subsets of F . Continuous linear functionalsare clearly bounded. The locally convex vector space of bounded linear operatorswith uniform convergence on bounded sets is denoted by L(E,F ), the dual spaceformed by bounded linear functionals by E′. These spaces are locally convex vectorspaces, which we shall need for analysis (see [24], 3.17).

Definition 12. Let E be a locally convex space, then c : R → E is calledsmooth if all derivatives exist as limits of difference quotients. The set of smoothcurves is denoted by C∞(R, E).

A subset U ⊂ E is called c∞-open if c−1(U) is open in R for all c ∈ C∞(R, E).The generated topology on E is called c∞-topology and E equipped with this topologyis denoted by c∞E.

If U is c∞-open, a map f : U ⊂ E → R is called smooth if f c ∈ C∞(R,R)for all c ∈ C∞(R, E).

These definitions work for any locally convex vector space, but for the followingtheorem we need a weak completeness assumption. A locally convex vector spaceE is called convenient if the following property holds: a curve c : R → E is smoothif and only if it is weakly smooth, i.e. l c ∈ C∞(R,R) for all l ∈ E′. This isequivalent to the assertion that any smooth curve c : R → E can be (Riemann-)integrated in E on compact intervals (see [24], 2.14). The spaces L(E,F ) and E′

are convenient vector spaces (see [24], 3.17), if E and F are convenient.

Theorem 17. Let E,G,H be convenient vector spaces, U ⊂ E, V ⊂ G c∞-open subsets:

(1) Smooth maps are continuous with respect to the c∞-topology.(2) Multilinear maps are smooth if and only if they are bounded.(3) If P : U → G is smooth, then dP : U → L(E,G) is smooth and bounded

linear in the second component, where

dP (f)h :=d

dt|t=0P (f + th).

(4) The chain rule holds.(5) Let [f, f + h] := f + sh for s ∈ [0, 1] ⊂ U , then Taylor’s formula is true

at f ∈ U , where higher derivatives are defined as usual (see iii.),

P (f + h) =n∑

i=0

1i!diP (f)h(i) +

∫ 1

0

(1− t)n

n!dn+1P (f + th) (h(n+1))dt

for all n ∈ N.


(6) There are natural convenient locally convex structures on C∞(U,F ) andwe have cartesian closedness

C∞(U × V,H) ' C∞(U,C∞(V,H)).

via the natural map f 7→ f : U → C∞(V,H) for f ∈ C∞(U ×V,H). Thisnatural map is well defined and a smooth linear isomorphism.

(7) The evaluation and the composition

ev : C∞(U,F )× U → F, (P, f) 7→ P (f)

. . : C∞(F,G)× C∞(U,F ) → C∞(U,G), (Q,R) 7→ Q Rare smooth maps.

(8) A map P : U ⊂ E → L(G,H) is smooth if and only if (evg P ) is smoothfor all g ∈ G.

Proof. For the proofs see [24] in Subsections 3.12, 3.13, 3.18, 5.11, 5.12,5.18.

Convenient Calculus is an extension of the Gateaux-Calculus to locally convexspaces, where all necessary tools for analysis are preserved. Since typically vectorspaces like C∞(U,F ) or L(E,F ) are not Frechet spaces, this extension is very usefulfor the analysis of the geometric objects in Section 3.

Theorem 18. Let E,F be Frechet spaces and U ⊂ E a c∞-open subset, thenU is open and P : U ⊂ E → F is Gateaux-smooth if and only if P is smooth (inthe convenient sense).

Concerning differential equations, there are possible counterexamples on non-normable Frechet spaces in all directions, which causes some problems in the foun-dations of differential geometry (see [24] and the review article [25]).

If not otherwise stated, E and F denote Frechet spaces and B a Banach spacein what follows. A vector field P on an open subset U ⊂ E is a smooth mapP : U → E. We denote by X(U) the convenient space of all vector fields on an opensubset of a Frechet space E. Given P : U ⊂ E → E a vector field on U . We arelooking for solutions of the ordinary differential equation with initial value g ∈ U

f :]− ε, ε[→ U smoothd

dtf(t) = P (f(t))

f(0) = g ∈ U.If for any initial value g in a small neighborhood V of f0 ∈ U there is a unique

smooth solution t 7→ fg(t) for t ∈]− ε, ε[ depending smoothly on the initial value g,then Fl(t, g) := fg(t) defines a local flow, i.e. a smooth map

Fl :]− ε, ε[×V → E

Fl(0, g) = g

F l(t, F l(s, g)) = Fl(s+ t, g)

if s, t, s + t ∈] − ε, ε[ and Fl(s, g) ∈ V . If there is a local flow around f0 ∈ U(this shall mean once and for all: “in an open, convex neighborhood of f0”), thedifferential equation is uniquely solvable around f0 ∈ U and the dependence oninitial values is smooth (see Lemma 6 for the proof). Notice at this point that it

3. GEOMETRY 33

is irrelevant if we define “smooth dependence” on initial values via smooth mapsV → C∞(] − ε, ε[, E) or V×] − ε, ε[→ E by cartesian closedness. We shall denotefg(t) = Flt(g) = Fl(t, g).

We can replace in the above definition of a local flow the interval ] − ε, ε[ by[0, ε[ to obtain local semiflows. The initial value problem

f : [0, ε[→ U smoothd

dtf(t) = P (f(t))

f(0) = g ∈ U.

admits unique solutions around an initial value depending smoothly on the initialvalues if and only if a local semiflow exists. Here we need convenient analysis ofnon-open domains, see [17] or [24]. The notion of a local semiflow is redundant onBanach spaces.

Lemma 6. Let Fl be a local semiflow on [0, ε[×U → E, then the map P (f) :=ddt |t=0Fl(t, f) is a well defined smooth vector field. We obtain

∂

∂tF lt(f)P (f) = P (Flt(f))

and the initial value problem has unique solutions for small times which coincidewith the given semiflow.

3. Geometry

We are interested in the geometry generated by a finite number of vector fieldsgiven on an open subset of a Frechet space E. Therefore we need the notions offinite-dimensional submanifolds (with boundary) of a Frechet space (see [24] forall details and more). Here and subsequent E denotes a Frechet space. From theclassical definition we can conclude the following Lemma (see [16]):

Lemma 7 (Submanifolds by Parametrization). Let E be a Frechet space andφ : U ⊂ Rn

≥0 → E a smooth immersion, i.e. for u ∈ U the map Dφ(u) is injective,then for any u0 ∈ U there is a small open neighborhood V of u0 such that φ(V ) isa submanifold with boundary of E, φ|V is a called a parametrization.

Definition 13 (Lie bracket). The Lie bracket of two vector fields X,Y ∈ X(U)is defined by the following formula:

[X,Y ](f) = dX(f) · Y (f)− dY (f) ·X(f)

and is a bounded, skew-symmetric bilinear map from X(U)× X(U) into X(U).

Definition 14. Let φ : U ⊂ Rn≥0 → E be a smooth parametrization of a

submanifold with boundary M ⊂ E, i.e. φ(U) = M. A vector field X : U → Rn

is called φ-related to Y : V ⊂ E → E if dφ(u) ·Xu = Yφ(u) for all u ∈ U . This isdenoted by X∼φY .

Proposition 2. Let U be an open set in E, and M⊂ U be a submanifold withboundary. If two vector fields X1, X2 ∈ X(U) are tangent to M, then [X,Y ](h) ∈ThM for h ∈ U .


Proof. Take a parametrization φ : U ⊂ Rn≥0 → E of M around h0 ∈ M.

By [24] we obtain, that if two vector fields are φ-related, then their Lie bracket isφ-related, too. Given two vector fields X1, X2 : U → E such that for all h ∈M wehave Xi(h) ∈ ThM for i = 1, 2. Then we can define vector fields Y1, Y2 on U byrestriction and pulling back to U such that

dφ(u) · Yi(u) = Xi(φ(u))

for i = 1, 2 and u ∈ U . So Xi is φ-related to Yi for i = 1, 2 and therefore their Liebracket as well. Consequently all Lie brackets take along M values in its tangentspace, since we can choose a parametrization around any point h0 ∈M.

From this observation Frobenius theory can be built up. We denote by 〈. . .〉the generated vector space over the reals R.

Definition 15. Let E be a Frechet space, U an open subset. A distribution onU is a collection of vector subspaces D = Dff∈U of E. A vector field X ∈ X(U)is said to take values in D if X(f) ∈ D(f) for f ∈ U . A distribution D on U issaid to be involutive if for any two locally given vector fields X,Y with values in Dthe Lie bracket [X,Y ] takes values in D.

A distribution is said to have constant rank on U if dimR Df is locally constantfor f ∈ U . A distribution is called smooth if there is a set S of locally defined vectorfields on U such that

Df = 〈X(f)|(X : UX → E) ∈ S and f ∈ UX〉.

We say that the distribution admits local frames on U if for any f ∈ U there is anopen neighborhood f ∈ V ⊂ U and n smooth, pointwise linearly independent vectorfields X1, ..., Xn on V with

〈X1(g), ..., Xn(g)〉 = Dg

for g ∈ V .

Remark 11. Given a distribution D on U generated by a set of local vectorfields S, such that the dimensions of Df are bounded by a fixed constant N . Letf ∈ U be a point with maximal dimension nf = dimR Df , then there are nf smoothlocal vector fields X1, ..., Xnf

∈ S with common domain of definition U ′ such that

〈X1(f), ..., Xn(f)〉 = Df .

Choosing nf continuous linear functionals l1, ..., lnf∈ E′ with li(Xj(f)) = δij, then

the continuous mapping M : U ′ → L(Rnf ), g 7→ (li(Xj(g))) has range in theinvertible matrices in a small neighborhood of f . Consequently in this neighborhoodthe dimension of Dg is at least nf . It follows by maximality of nf that it is exactlynf . In particular the distribution admits a local frame at f .

The concept of weak foliations will be perfectly adapted to the FDR-problem:

Definition 16. A weak foliation F of dimension n on an open subset U of aFrechet space E is a collection of submanifolds with boundary Mrr∈U such that

(1) For all r ∈ U we have r ∈Mr and the dimension of Mr is n.(2) The distribution

D(F)(f) := 〈TfMr for all r ∈ U with f ∈Mr〉

4. SYNTHESIS 35

has dimension n for all f ∈ U , i.e. given f ∈ U the tangent spaces TfMr

agree for all Mr 3 f . This distribution is called the tangent distributionof F .

Given any distribution D we say that D is tangent to F if D(f) ⊂ D(F)(f)for all f ∈ U .

Theorem 19. Let D be an smooth distribution of constant rank n on an opensubset U of a Frechet space E. Assume that for any point f0 the distribution admitsa local frame of vector fields X1, ..., Xn, where X1, ..., Xn−1 admit local flows FlXi

t

and Xn admits a local semiflow FlXnt . Then D is involutive if and only if it is

tangent to an n-dimensional weak foliation.

For details on Frobenius theorems in the classical setting see [23]. The phe-nomenon that there is no Frobenius chart is due to the fact that there is one vectorfield among the vector fields X1,...,Xn (generating the distribution D) admittingonly a local semiflow. If all of them admitted flows, there would exist a Frobeniuschart, which can be given by a construction outlined in [34]. The non-existence ofa Frobenius-chart means that the leafs cannot be parallelized, since they follow asemiflow, which means in turn that ”gaps” between two leafs can occur and leafscan touch. This is an infinite dimensional phenomenon, which does not appear infinite dimensions.

4. Synthesis

From the geometric considerations we can conclude the existence of FDRs if

1: the distribution DLA generated by α, σ1, . . . , σd and all mutually gener-ated Lie brackets has locally constant dimension NLA ≥ 1.

2: there is a frame of vector fields X1, . . . , XNLAfor DLA around any point,

such that X1, . . . , XNLA−1 admit local flows and XNLAadmits a local

semiflow.

Applying Theorem 19 then yields the existence of NLA-dimensional locallyinvariant submanifolds with boundary. For an interesting application one has toinvestigate the meaning of 1. closely and guarantee 2. in as many situations aspossible. In this subsection the conditions (A1)–(A3) are applied such that onlythe algebraic condition remains and one can nicely distinguish between the vectorfields admitting flows and the one only admitting a semiflow.

Definition 17. Given a Frechet space E, a smooth map P : U ⊂ E → E iscalled a Banach map if there are smooth (not necessarily linear) maps R : U ⊂E → B and Q : V ⊂ B → E such that P = Q R, where B is a Banach space andV ⊂ B is an open set.

We denote by B(U) the set of Banach map vector fields on an open subset of aFrechet space E.

Theorem 20. B(U) is a C∞(U,R)-submodule of X(U).

Proof. We have to show that for ψ, η ∈ C∞(U,R) and P1, P2 ∈ B(U) thelinear combination ψP1 + ηP2 ∈ B(U). Given Pi = Qi Ri for i = 1, 2 withintermediate Banach spaces Bi, then ψP1 + ηP2 = Q R with Q : R2 × V1 × V2 ⊂


R2 ×B1 ×B2 → E and R : U → R2 ×B1 ×B2 such that

Q(r, s, v1, v2) = rQ1(v1) + sQ2(v2)

R(f) = (ψ(f), η(f), R1(f), R2(f))

So the sum ψP1 + ηP2 is a Banach map and therefore the set of all Banach mapvector fields carries the asserted submodule structure.

Lemma 8. Let U be an open set in a Frechet space E, then B(U) is a subalgebrawith respect to the Lie bracket. Let A be a bounded linear operator on E, then[A,B(U)] ⊂ B(U). Consequently the Lie algebra L(E) acts on B(U) by the Liebracket.

Proof. Given two Banach maps P1 and P2, dP1(f) · P2(f) = dQ1(R1(f)) ·dR1(f) ·P2(f) holds, which can be written as composition of dQ1(v) ·w for v, w ∈ Band (R1(f), dR1(f) · P2(f)) for f ∈ U . So the Lie bracket lies in B(U). GivenA ∈ L(E), we see that AP1(f) − dP1(f) · Af is a Banach map by an obviousdecomposition.

Banach map vector fields admit solutions of initial value problems.

Theorem 21 (Banach map principle). Let P : U ⊂ E → E be a Banach map,then P admits a local flow around any point g ∈ U .

Proof. For the proof see [20], Theorem 5.6.3.

We are in particular interested in special types of differential equations onFrechet spaces E, namely Banach map perturbed bounded linear equations. Givena bounded linear operator A : E → E, the abstract Cauchy problem associated toA is given by the initial value problem associated to A. We assume that there is asmooth semigroup of bounded linear operators S : R≥0 → L(E,E) such that

limt↓0

St − id

t= A

which is a global semiflow for the linear vector field f 7→ Af . Notice that thetheory of bounded linear operators on Frechet spaces contains as a special caseHille-Yosida-Theory of unbounded operators on Banach spaces (see for example[33]).

Given a strongly continuous semigroup St for t ≥ 0 of bounded linear operatorson a Banach space B, then D(An) with the respective operator norms pn(f) :=∑n

i=0 ||Aif || for n ≥ 0 and f ∈ D(An) is a Banach space, where the semigrouprestricts to a strongly continuous semigroup S(n). Consequently the semigrouprestricts to the Frechet space D(A∞). This semigroup is now smooth.

Given a Banach map P : U ⊂ E → E, we want to investigate the solutions ofthe initial value problem

d

dtf(t) = Af(t) + P (f(t)), f(0) = f0.

Theorem 22. Let E be a Frechet space and A be the generator of a smoothsemigroup S : R → L(E) of bounded linear operators on E. Let P : U ⊂ E → E

5. EXAMPLES 37

be a Banach map. For any f0 ∈ U there is ε > 0 and an open set V containing f0and a local semiflow Fl : [0, ε[×V → U satisfying

d

dtF l(t, f) = AFl(t, f) + P (Fl(t, f))

Fl(0, f) = f

for all (t, f) ∈ [0, ε[×V .

Proof. For the proof see [16].

For the purposes of classification we shall need the following result, see [17].

Lemma 9. Let A be the generator of a strongly continuous semigroup S on aBanach space B, then the operator A : D(A∞) → D(A∞) is a Banach map if andonly if A : B → B is bounded.

Now we can formulate the following conclusions from (A1)–(A3), which makesthe geometric conditions applicable:

1: The vector fields σi and α are Banach map vector fields for i = 1, . . . , n.2: The Lie brackets [α, σi] and [σi, σj ] are Banach maps for i, j = 1, . . . , n.

Any further Lie bracket with a Banach map vector field yields a Banachmap vector field. This is due to Lemma 8.

3: The vector field µ is not a Banach map due to (A3) and Lemma 9, butgenerates a local semiflow due to Theorem 22.

Theorem 23. Let U ⊂ H be a domain of definition in a separable Hilbert spaceH, where (A1)–(A3) are in force. We denote V = U ∩D(A∞), where we expect tofind tangent leaves for the given vector fields. We define

Dr := 〈µ(r), σ1(r), . . . , σd(r), [µ, σi](r), [σi, σj ](r), . . . 〉for r ∈ V and assume that there is N ≥ 1 such that dimDr = N for r ∈ V .Then there exists a weak foliation on V such that D is its tangent distribution.Furthermore if the volatility structure is non-trivial, then N ≥ 2 and µ is linearlyindependent of all other Lie brackets.

Proof. See [18] for the proof.

We can also consider the question of possible singular sets, i.e. points whereµ and the other Lie brackets are linearly dependent. A global analysis yields (see[18]) that those points form a nowhere dense set. In financial mathematics thosepoints correspond to time-autonomous interest rate models (see the next section).

5. Examples

Here we assume that A = ddx and H is an appropriate Hilbert space of forward

curves. We let (A1)–(A3) be in force on V = U ∩D(A∞). Hence we have that

σ(r) = Φ(r)λ, r ∈ Ufor some λ ∈ D(A∞0 ) \ 0 and a smooth map Φ : U → R (which is of the formΦ = φ ` by (A1)). Without loss of generality we can assume that Φ > 0, since(A1) requires “linear independence” of Φ which here simply means Φ 6= 0.

In the seminal papers [6] and [4] finite-dimensional realizations, in particularthe Hull-White extensions of the Vasicek and CIR-model, are considered for the


first time from the geometric point of view. In addition to their excellent treat-ment (compare Section 5 of [4] or Section 7 of [6]), we prove that the Hull-Whiteextensions of the Vasicek and CIR model are essentially the only 2-dimensional lo-cal HJM models and we demonstrate the importance of the corresponding singularsets.

We want to specify under which conditions this volatility structure admits 2-dimensional realizations and how they look like. We shall show that it has to beeither of the Vasicek or CIR type. This is already done in Section 7.3 of [6], however,their special setting does not allow to treat the CIR-case.

Writing ψ(r) := Φ(r)(DΦ(r) · λ), we obtain for r ∈ U ∩D(A∞)

dσ(r) · h = (dΦ(r) · h)λdσ(r) · σ(r) = Φ(r)(dΦ(r) · λ)λ = ψ(r)λ

µ(r) =d

dxr + Φ(r)2λ

∫λ− 1

2ψ(r)λ

dµ(r) · h =d

dxh+ 2Φ(r)(dΦ(r) · h)λ

∫λ− 1

2(dψ(r) · h)λ.

Consequently we can calculate the Lie bracket

[µ, σ](r) = Φ(r)d

dxλ+ 2Φ(r)ψ(r)λ

∫λ− 1

2Φ(r)(dψ(r) · λ)λ−

− (dΦ(r) · ddxr)λ− Φ(r)2(dΦ(r) · λ

∫λ)λ+

12ψ(r)(dΦ(r) · λ)λ.

We necessarily have [µ, σ](r) ∈ 〈λ〉 on U ∩ D(A∞). We can divide by Φ(r) andobtain an equation

d

dxλ+ 2ψ(r)λ

∫λ− θ(r)λ = 0

with some smooth function θ : U ∩D(A∞) → R. There are consequently two cases:(1) If λ and λ

∫λ are linearly independent in D(A∞), then by derivation

with respect to r we obtain that ψ and θ are constant, say 2ψ(r) = aand θ(r) = b with real numbers a and b. Defining Λ :=

∫λ we obtain

finally a Riccati equation for Λ, which yields the CIR-type if a 6= 0 or theVasicek-type if a = 0:

(3.1)d

dxΛ +

a

2Λ2 + bΛ = λ(0), Λ(0) = 0.

The Ho-Lee model is considered as particular case of the Vasicek modelfor b = 0.

(2) If λ and λ∫λ are linearly dependent inD(A∞), then we necessarily obtain

an equation of the type

d

dxλ+ bλ = 0,

which yields that λ is vanishes identically, since otherwise λ and λ∫λ are

linearly independent. This case was excluded at the beginning.Notice that λ(0) = 0 if and only if λ = 0, which is not possible. Hence a fortiori

we have λ(0) 6= 0, such that by rescaling we always can assume that λ(0) = 1. Thisobservation slightly improves the discussion in Section 7.3 in [6].

5. EXAMPLES 39

By the definition of ψ we have dΦ2(r) · λ = a, hence we obtain the followingrepresentation for Φ. We split D(A∞) into Rλ+E, where E := ker ev0. We denoteby pr : D(A∞) → E the corresponding projection. Then

(3.2) Φ(h) =√aev0(h) + η(pr(h)),

where η : pr(U∩D(A∞)) ⊂ E → R is a smooth function (compare with Proposition7.3 of [6]).

We have

S =h ∈ U ∩D(A∞) | π(h) = Ah+ Φ(h)2λ

∫λ ∈ 〈λ〉

.

Thus, if λ and λ∫λ are linearly independent inD(A∞) then any h ∈ S is necessarily

of the form

h = a1 + a2Λ2 + a3Λ

in all cases for some real numbers ai. By the particular representation of Φ weobtain that a2 = aa1 + g(a3), where g is some smooth real function derived from

aa1 + η(a2Λ2 + a3Λ) = a2.

By FlX we denote the local (semi-)flow of a vector field X on U ∩D(A∞). Theleaves through r∗ of the weak foliation are given by the local parametrization

(u0, u1) 7→ Flπu0(r∗) + u1

d

dxΛ

if r∗ does not lie in the singular set S. If r∗ ∈ S, then the leaf is a one dimensionalimmersed submanifold of 〈1,Λ,Λ2〉. Notice that the stochastic evolution of thefactor process takes place in the u1-component.

We summarize the preceding results in the following theorem.

Theorem 24. Let S and U be as above. Assume that Σ admits a 2-dimensionalrealization around any initial curve r∗ ∈ U ∩ D(A∞) \ S. Then there exists λ ∈D(A∞0 ) and a function Φ : U → R>0 such that σ(h) = Φ(h)λ. The singular set Sis a (possibly empty) subset of 〈1,Λ,Λ2〉, where Λ =

∫λ satisfies a Riccati equation.

The local HJM model is an affine short rate model. That is, for every initial curver∗ ∈ U ∩D(A∞) there exist functions b : R≥0×R → R, θ : R≥0 → R and a stoppingtime τ > 0 such that

(3.3) rt∧τ = Flµt∧τ (r∗) +Rt∧τ λ

is the unique U -valued local solution and the short rates Rt = rt(0) follow, locallyfor t ≤ τ , a time-inhomogeneous diffusion process

dRt = b(t, Rt) dt+√aRt + θ(t) dWt.

This process becomes time-homogeneous if and only if r∗ ∈ S, and then rt∧τ ∈ Sfor all t ≥ 0.

Proof. We know that λ(0) 6= 0.


5.1. The Hull-White extension of the Vasicek model. We consider thevolatility structure of the Vasicek model: σ(r)(x) = ρ exp(−βx) = ρλ with ρ > 0and β > 0, for r ∈ U ∩D(A∞) = D(A∞) and x ≥ 0. Then by the above formulas

[µ, σ] = −βρλ.

The singular set S is characterized by

d

dxh+

ρ2

βexp(−βx)(1− exp(−βx)) = c exp(−βx)

for some real c. Therefore a2 is some fixed value, namely a2 = ρ2

2 and a1, a3 arearbitrary. Consequently the singular S set is an affine subspace for the fixed valuesρ, β:

h = a1 −ρ2

2Λ2 + a3Λ.

Going back to traditional notations for the Vasicek model we write

Λ(x) =1β

(1− exp(−βx))

BV (x) = Λ′(x) = e−βx

AV (x) = bΛ(x)− ρ2

2Λ(x)2,

then h lies in the singular set S if and only if

h ∈ AV + 〈BV 〉

for some value b (which becomes an additional parameter in the short rate equation).The solution for r∗ in the singular set reads as follows

rt = AV +BV Rt

dRt = (b− βRt) dt+ ρ dWt,

where Rt = ev0(rt) denotes the short rate, which is the Vasicek short rate model.Outside the singular set S we have a 2-dimensional realization. First we cal-

culate the deterministic part of the dynamics

Flµu0(r∗)(x) = Su0r

∗(x) +∫ u0

0

Su0−s(ρ2

βexp(−βx)(1− exp(−βx)) ds

= Su0r∗(x) +

ρ2

2

∫ u0

0

d

dx(Λ)2(x+ u0 − s) ds

= r∗(x+ u0) +ρ2

2Λ(x+ u0)2 −

ρ2

2Λ(x)2.

If we identify u0 with the time variable t, which is possible since the stochasticsonly occurs in direction of BV ,

rt(x) = r∗(x+ t) +ρ2

2Λ(x+ t)2 − ρ2

2Λ(x)2 + Λ′(x)Zt

dZt = −βZt dt+ ρ dWt.

A parameter transformation yields the customary form, namely

Rt = e−βtr∗(0) +∫ t

0

e−β(t−s)b(s) ds+ Zt.

5. EXAMPLES 41

This yields the following expressions:

AHWV (t, x) = r∗(x+ t) +ρ2

2Λ(x+ t)2 − ρ2

2Λ(x)2−

− (Λ′(x))2r∗(0)− Λ′(x)∫ t

0

e−β(t−s)b(s) ds

BHWV (x) = BV (x) = Λ′(x)

dRt = (b(t)− βRt) dt+ ρ dWt

rt = AHWV (t) +BHWV Rt

b(t) =d

dtr∗(t) + βr∗(t) +

ρ2

2β(1− exp(−2βt)).

The functions AHWV and BHWV are solutions of time-dependent Riccati equationsconstructed by geometric methods. The equation for b follows from the fact thatAHWV (t, 0) = 0.

5.2. The Hull-White extension of the CIR model. We proceed in thesame spirit: σ(r) := ρ

√ev0(r)λ for ρ > 0. The volatility structure is defined on the

convex open set U = ev0(r) > ε for some ε > 0. The function Λ :=∫λ satisfies

(in certain normalization) a Riccati equation, namely

d

dxΛ +

ρ2

2Λ2 + βΛ = 1, Λ(0) = 0.

We obtain the solution (see e.g. [13, Section 7.4.1])

Λ(x) =2 exp(x

√β2 + 2ρ2)− 1

(√β2 + 2ρ2 − β)(exp(x

√β2 + 2ρ2)− 1) + 2

√β2 + 2ρ2

.

Under this assumption we can proceed as above: the singular set S is determinedby the equation

µ(h) =d

dxh+ ρ2ev0(h)ΛΛ′ ∈ 〈λ〉 ,

hence

h = a1 +ρ2

2a1Λ2 + a3Λ.

Again a1 and a3 can be chosen freely, which completely determines S. Traditionallyone writes the singular set in the following form:

ACIR = bΛ

BCIR = 1− βΛ− ρ2

2Λ2 = Λ′

with some additional parameter b and we obtain equally that h lies in S if and onlyif

h ∈ ACIR + 〈BCIR〉 .The short rate dynamics follows the known pattern:

rt = ACIR +BCIRRt

dRt = (b− βRt) dt+ ρ√Rt dWt


for r∗ ∈ S. Outside the singular set we have a 2-dimensional realization. First wecalculate the deterministic part, by the variation of constants formula,

Flµu0(r∗)(x) = Su0r

∗(x) + ρ2

∫ u0

0

FlΠs (r∗)(0)(Su0−s(Λ′Λ))(x) ds.

Identifying u0 with the time parameter yields the following formula 2-dimensionalrealization, which is derived by direct calculations,

rt = Flµt (r∗) + Λ′Zt

dZt = −βZt dt+ ρ√c(t) + Zt dWt,

where c(t) = FlΠt (r∗)(0). The short rate is given through Rt = c(t) + Zt and

dRt = (βc′(t)− βZt) dt+ ρ√Rt dWt

= (b(t)− βRt) dt+ ρ√Rt dWt.

Notice that λ(0) = Λ′(0) = 1 by the Riccati equation and b(t) = c′(t) + βc(t).This formula closes the circle with the classical Hull-White extension of the

CIR-model:

AHWCIR(t, x) = Flµt (r∗)(x)− c(t)Λ′(x)

BHWCIR = BCIR = Λ′

dRt = (b(t)− βRt) dt+ ρ√Rt dWt

rt = AHWCIR(t) +BHWCIRRt

b(t) = βc(t) +d

dtc(t)

c(t) = r∗(t) + ρ2

∫ t

0

c(s)(ΛΛ′)(t− s) ds.

Again this is a geometrical construction of solutions of time-dependent Riccatiequations.

CHAPTER 4

Malliavin Calculus in infinite dimensions

1. Gaussian Spaces and Malliavin Derivatives

Gaussian probability spaces are the playground of Malliavin Calculus:

Definition 18. A probability space (Ω,F , P ) is called Gaussian if there ex-ists a closed, separable subspace G ⊂ L2(Ω,F , P ) of normally distributed, centeredrandom variables and

F = σ(X for X ∈ G).

Definition 19. Given a real, separable Hilbert space H and a probability space(Ω,F , P ), then an isometry η : H → L2(Ω,F , P ), i.e η is linear and

E(η(h)η(g)) = 〈h, g〉

for all h, g ∈ H, is called an isonormal Gaussian process. We shall always assumethat F = σ(η(h) for h ∈ H). Hence the probability space (Ω,F , P ) together withthe subspace η(H) is Gaussian.

Example 5. Take a centered Gaussian process (Xt)0≤t≤1 on (Ω,F , P ) withcovariance function R(s, t) := cov(XtXs). We assume that (Xt)0≤t≤1 generatesthe σ-Algebra F . For instance, R(s, t) = s ∧ t for Brownian motion, or R(s, t) =12 (s2h+t2h+|t−s|2h) for fractional Brownian motion with Hurst parameter h ∈]0, 1[(for h = 1

2 we obtain Brownian motion). The Reproducing Kernel Hilbert space His defined as the completion of the pre-Hilbert space E =

⟨1[0,t] for t ∈ [0, 1]

⟩under

the scalar product⟨1[0,s], 1[0,t]

⟩:= R(s, t). We define η : H → L2(Ω,F , P ) via

η(1[0,t]) = Xt and obtain an isometry to the closure of 〈Xt for t ∈ [0, 1]〉.

If an isonormal Gaussian process exists, we are given a Gaussian probabilityspace and if we are given a Gaussian space one can specify an isonormal Gaussianprocess as sort of coordinates. On a Gaussian space (Ω,F , P ) with G ⊂ L2(Ω,F , P )together with an isonormal Gaussian process η : H → L2(Ω,F , P ), we shall alwaysassume that the two structures are compatible, i.e. G = η(H). Many properties donot depend on η but only on G.

Proposition 3. Let (Ω,F , P ) be a Gaussian space with isonormal process η,then the linear hull of eη(h), h ∈ H is dense in L2(Ω,F , P ).

Proof. We take a ONB (hi)i≥1 and define normally distributed N(0, 1), in-dependent random variables Xi := η(hi) for i ≥ 1. We can proceed by conditionalexpectations and characteristic functions.

We shall apply the following classes of functions on Rn, let

C∞0 (Rn) ⊂ C∞b (Rn) ⊂ C∞p (Rn)

43

44 4. MALLIAVIN CALCULUS IN INFINITE DIMENSIONS

denote the functions with compact support, with bounded derivatives of all ordersand with derivatives of polynomial growth. We introduce random variables of theform

F := f(η(h1), ..., η(hn))

for hi ∈ H. If f belongs to one of the above classes of functions, the associatedrandom variables are denoted by

S0 ⊂ Sb ⊂ Sp.

We speak of smooth random variables. The polynomials of elements η(h) are de-noted by P. By Proposition 3 the vector space P is dense in L2(Ω,F , P ). Sincewe can approximate functions with compact support uniformly by polynomials, thevector space S0 is dense, too.

The representation of a smooth random variable as a smooth functional isunique in the following sense:

F = f(η(h1), ..., η(hn))

= g(η(g1), ..., η(gm)),

then we generate the linear space 〈h1, .., hn, g1, ..., gn〉 with orthonormal basis(ei)1≤i≤k and representation

hi =k∑

l=1

ailel

gj =k∑

l=1

bjlel.

Then the functions f A and g B coincide, since

(f A)(η(e1), ..., η(ek)) = (g B)(η(e1), ..., η(ek)),

the random variables (η(e1), . . . , η(ek)) admits a density on Rk, and f, g are con-tinuous.

Definition 20. Let F ∈ Sp. The Malliavin derivative DF ∈ L2(Ω,F , P )⊗His defined by the following formula

F = f(η(h1), ..., η(hn))

DF :=n∑

i=1

∂

∂xif(η(h1), ..., η(hn))hi.

If we are given a concrete representation H = L2(T,B, µ), then we identify

L2(Ω,F , P )⊗H = L2(Ω× T,F ⊗ B, P ⊗ µ)

and we obtain a measurable process (DtF )t∈T as Malliavin derivative.

1. GAUSSIAN SPACES AND MALLIAVIN DERIVATIVES 45

The Malliavin derivative is well defined, i.e. it does not depend on the repre-sentation of F as smooth random variable by the previous uniqueness assertion,

n∑i=1

∂

∂xif(η(h1), ..., η(hn))hi =

k∑j=1

∂

∂yj(f A)(η(e1), ..., η(en))ej

=k∑

j=1

∂

∂yj(g B)(η(e1), ..., η(en))ej

=m∑

i=1

∂

∂xig(η(g1), ..., η(gm))hi

due to the chain rule. We have furthermore the following idea of directional deriv-ative, which shall be discussed later.

〈DF, h〉H =d

dtf(η(h1) + t 〈h1, h〉 , ..., η(hn) + t 〈hn, h〉)|t=0

for h ∈ H.We shall now prove the first partial integration result, which is of fundamental

importance:

Proposition 4. Let F be a smooth random variable and h ∈ H, then

E(〈DF, h〉) = E(Fη(h)).

Proof. The equation in question can be normalized such that ||h|| = 1. Thereare by a transformation of variables orthonormal elements ei such that

F = f(η(e1), ..., η(en))

with f ∈ C∞p (Rn) and h = e1. Then

E(〈DF, h〉) =∫

Rn

∂f

∂x1(x)

1√(2π)n

exp(−||x||2

2)dx

partial integration=

∫Rn

f(x)x11√

(2π)nexp(−||x||

2

2)dx

= E(Fη(e1)) = E(Fη(h)).

Lemma 10 (partial integration). Suppose η is an isonormal process on H andF,G ∈ S, then for h ∈ H

E(G 〈DF, h〉) + E(F 〈DG,h〉) = E(FGη(h)).

Proof. The proof is clear from the above lemma and by the fact that

D(FG) = FDG+GDF

for F,G ∈ Sp.

We have already definedD : S ⊂ Lq((Ω,F , P )) → Lq((Ω,F , P );H) = Lq((Ω,F , P ))⊗H for q ≥ 1, and we see that this operator is closeable by partial integration: given


a sequence of smooth functionals Fn → 0 in Lq and DFn → G in Lq(Ω,F , P ;H)as n→∞, then

E(〈G, h〉H F ) = limn→∞

E(〈DFn, h〉F ) =

= limn→∞

E(−Fn 〈DF, h〉) + limn→∞

E(FnFηh) = 0

for F ∈ Sp. Notice that Sp ⊂ ∩q≥1Lq. So G = 0 and therefore D is closeable. We

denote the closure on each space by D1,q.

Definition 21. Given q ≥ 1, then we denote by

||F ||1,q := (E(|F |q) + E(||DF ||qH))1q

the operator norm for any F ∈ S. By closeability we know that the closure of thisspace is a Banach space, denoted by D1,q and a Hilbert space for q = 2. We havethe continuous inclusion

D1,q → Lq((Ω,F , P ))

which has as image the maximal domain of definition of D1,q in Lq, where we shallwrite - by a small abuse of notation - D for the Malliavin derivative.

By tensoring the whole procedure we can define Malliavin derivative for smoothfunctionals with values in V , an additionally given Hilbert space,

S ⊗ V ⊂ Lp((Ω,F , P ))⊗ V,

where we take the algebraic tensor products. We define the Malliavin derivative onthis space by D⊗id aknowledging that analysis usually does not depend on the rangespace and extend it to Lq(Ω,F , P ;V ). So we get for F (ω) = (f ⊗ v)(ω) = f(ω)vthe Malliavin derivative

DF = Df ⊗ v.

Consequently we can define higher derivatives via iteration

DkF = DDk−1F

for smooth functionals F ∈ Lq(Ω,F , P ) ⊗ V . Closing the spaces we get Malliavinderivatives Dk on Lq(Ω,F , P ;V ) to Lq((Ω,F , P );V ⊗ H⊗k), which are closeableoperators by induction. We define the norms (Hilbert norms)

||F ||k,q := (E(|F |q) +k∑

j=1

E(||DjF ||qV⊗H⊗j ))1q

for k ≥ 1 and q ≥ 1. The respective closed spaces Dk,q(V ) are Banach spaces(Hilbert spaces), the maximal domains of Dk in Lq(Ω,F , P ;V ). The Frechet space∩p≥1 ∩k≥1 Dk,p(V ) is denoted by D∞(V ).

We see immediately the monotonicity

||F ||k,p ≤ ||F ||j,qfor p ≤ q and k ≤ j by norm inequalities of the type

||f ||p ≤ ||f ||qfor 1 ≤ p ≤ q for f ∈ ∩p≥1L

p(Ω,F , P ).Next we consider chain rules for the Malliavin derivative:

1. GAUSSIAN SPACES AND MALLIAVIN DERIVATIVES 47

Proposition 5. Let φ ∈ C1b (Rn) be given, such that the partial derivatives are

bounded and fix p ≥ 1. If F ∈ D1,p(Rn), then φ F ∈ D1,p and

D(φ F ) =n∑

i=1

∂φ

∂xi(F )DF i

which is the classical chain rule. Therefore D∞ is an C∞-algebra.

Proof. The proof is made by approximating F i by smooth variables F in and φ

by φ∗ψε, where ψε is a Dirac sequence of smooth functions. For the approximatingterms the formula is satisfied, then we obtain

||n∑

i=1

∂φ

∂xi(F )DF i −D((φ ∗ ψε) F i

n||p → 0

as ε→ 0 and n→∞, so by closedness we obtain since (φ ∗ ψε) F in → φ F in Lp

as ε→ 0 and n→∞ the result.

For the next generalization of the chain rule we need the follwing technicalresult: Given a sequence Fm → F in L2(Ω,F , P ) with Fm ∈ D1,2 and

supmE(||DFm||2H) <∞,

then F belongs to D1,2 and the sequence of derivatives DFm → DF as m→∞ inthe weak topology of L2(Ω,F , P ;H), i.e.

E(〈DFm, h〉H G) → E(〈DF, h〉H G)

as m → ∞, for all h ∈ H and G ∈ S0. This follows immediately from partialintegration, since

E(〈DFm, h〉H G) = E(〈D(FmG), h〉H)− E(〈D(G), h〉H Fm)

= E(Fmη(h))− E(〈D(G), h〉H Fm)

→m→∞ E(FGη(h))− E(〈D(G), h〉H Fm)

= E(〈DF, h〉H G).

Since supmE(||DFm||2H) <∞, we can extend the convergence ofE(〈DFm, h〉H G) →E(〈DF, h〉H G), to weak convergence.

Theorem 25. Let φ be a given global Lipschitz function with constant K > 0.If F ∈ D1,2(Rn), then φ F ∈ D1,2 and there exists a random vector (G1, ..., Gn)bounded by K such that

D(φ F ) =n∑

i=1

GiDFi.

Proof. We redo the the proof with regularizations: We choose φn smoothconverging to φ uniformly on compacts with derivatives bounded by K. Thenthe sequence φn(F ) converges to φ(F ) in L2 and the sequence D(φn F )n≥0 isbounded in L2(Ω,F , P ;H). So φ F lies in D1,2 and the limit is given in the weaktopology. Since ∇φn F is bounded by K, it is bounded in L2(Ω,F , P ; Rn), sowe obtain weak convergence to a random vector (G1, ..., Gn) bounded by K of asubsequence. If we take the weak limit in

D(φn(F )) =n∑

i=1

∂φn

∂xi(F )DF i


we arrive at the result.

2. The Skorohod Integral and Partial Integration

Let (Ω,F , P ) be a Gaussian space with isonormal Gaussian process η : H →L2(Ω,F , P ), the Malliavin derivative is denoted by

D : D1,2 ⊂ L2(Ω) → L2(Ω)⊗H = L2(Ω;H).

The adjoint operator δ : dom1,p(δ) ⊂ Lp(Ω) ⊗ H → L2(Ω) is a closed denselydefined operator for 1 < p < ∞. We are particularly interested in the case p = 2.By definition u ∈ dom1,2(δ) if and only if F 7→ E(〈DF, u〉) for F ∈ D1,2 is a boundedlinear functional on L2(Ω). If u ∈ dom1,2(δ), we have the following fundamentalpartial integration formula

E(〈DF, u〉) = E(Fδ(u))

for F ∈ D1,2. δ is called the Skorohod integral or divergence operator. By thepartial integration result of Proposition 4 we obtain immediately H ⊂ dom1,2(δ),the deterministic strategies, with δ(1⊗ h) = η(h).

A smooth elementary process is given by

u =n∑

j=1

Fj ⊗ hj

with Fj ∈ Sp and hj ∈ H. We shall denote the set of such processes by the(algebraic) tensor product Sp ⊗H. By integration by parts we can conclude thatSp ⊗H ⊂ dom1,2(δ) and

δ(u) =n∑

j=1

Fjη(hj)−n∑

j=1

〈DFj , hj〉H ,

since

E(〈u,DG〉H) =n∑

j=1

E(Fj 〈hj , DG〉H)

=n∑

j=1

E(〈hj , DFjG〉H)− E(G 〈hj , DFj〉H)

=n∑

j=1

E(Fjη(hj)G)− E(G 〈hj , DFj〉H).

For the most important conclusion on the Skorohod integral, namely the integra-bility of progressively measurable strategies, we need the following Lemma.

Lemma 11. Given a Gaussian probability space with isonormal Gaussian pro-cess η and define a sub-σ-algebra G ⊂ F with a by means of a closed subspaceG ⊂ H via

G := σ(η(h) for h ∈ G).If F ∈ D1,2 is G-measurable, then

〈h,DF 〉H = 0

P -almost surely, for all h ⊥ G.

3. EXISTENCE OF DENSITIES 49

Proof. The almost sure identity holds if F = f(η(h1), . . . , η(hn)) with f ∈ Sp,for hi ∈ G by the very definition of the Malliavin derivative. Now, every F ∈ D1,2

can be approximated by such smooth random variables in L2, hence a subsequenceconverges almost surely to 〈h,DF 〉H , which vanishes then a fortiori.

Given a d-dimensional Brownian motion (Bt)t≥0 in its natural filtration (Ft)t≥0,then (Ω,F , P ) is a Gaussian space with isonormal Gaussian process

η(h) :=d∑

k=1

∫ ∞

0

hk(s)dBks

for h ∈ H := L2(R≥0,Rd). Define Gt ⊂ H those functions with support in [0, t] fort ≥ 0, then

Gt = σ(η(h) for h ∈ Gt) = Ft

by construction. Hence for F ∈ D1,2, which is Gt-measurable, we obtain that1[0,t]DF = DF almost surely. Consequently we obtain for a simple, predictablestrategy

u(s) =n∑

j=1

Fj ⊗ hj

with hj = 1]tj ,tj+1]ek, for 0 = t0 < t1 < · · · < tn+1 and Fj ∈ L2(Ω,Ftj, P ) for

j = 1, . . . , n, and ek ∈ Rd a canonical basis vector, that

δ(u) =n∑

j=1

Fjη(hj)−n∑

j=1

〈DFj , hj〉H

=n∑

j=1

Fj(Bkj+1 −Bk

j ).

Hence we can state the following theorem:

Theorem 26. Given a progressively measurable strategy u ∈ L2prog(Ω×R≥0; Rd),

then

δ(u) =d∑

k=1

∫ ∞

0

uk(s)dBks .

Proof. The Skorohod integral is a closed operator, the Ito integral is contin-uous on the space of progressively measurable strategies. Both operators coincideon the dense subspace of simple predictable strategies, hence – by the fact that δis closed – we obtain that they conincide on L2

prog(Ω× R≥0; Rd).

3. Existence of Densities

In this chapter the most famous early application of Malliavin Calculus istreated, the probabilistic proof of Hormander’s sum of the squares Theorem. There-fore we analyse first absolute continuity of laws of random variables on Wienerspace. We shall apply the notion of local random variables F ∈ D1,p

loc for p ≥ 1, i.e.there exists a sequence (Ωn, Fn)n≥0, where Ωn is a measurable set and Fn ∈ D1,p

for n ≥ 0 such that

Ωn ↑ Ω almost surely,Fn1Ωn

= F1Ωnalmost surely.


We denote by η : H → L2(Ω,F , P ) an isonormal Gaussian process. We assumethat F is generated by η. We illustrate the approach by a ”one-dimensional”example:

Proposition 6. Let F be random variable in D1,2 and suppose that DF||DF ||2H

is Skorohod integrable. Then the law of F has a continuous and bounded density fwith respect to the Lebesgue measure λ given by

f(x) = E

[1F>xδ

(DF

||DF ||2H

)]for real x.

Proof. We consider ψ(y) = 1[a,b] for a < b and φ(y) :=∫ y

−∞ ψ(x)dx. Sinceφ F ∈ D1,2 we obtain

〈D(φ F ), DF 〉H = ψ F ||DF ||2H

which allows to compute ψ(F ). We apply the duality relation between D and δand arrive at

E(ψ F ) = E

(⟨D(φ F ),

DF

||DF ||2H

⟩H

)=

= E

(⟨φ F, δ

(DF

||DF ||2H

)⟩H

)which leads to

P (a ≤ F ≤ b) = E

(⟨∫ F

−∞ψ(x)dx, δ

(DF

||DF ||2H

)⟩H

)=

=∫ b

a

E

(⟨1F>x, δ

(DF

||DF ||2H

)⟩H

)dx

by Fubini’s theorem.

In this theorem already all the ideas are contained, which make the ”partial in-tegration approach” (duality relation) successful. For RN -valued processes we needan additional technical Lemma. It is based on the Gagliardo-Nirenberg inequalitysaying that

||f ||L

NN−1

≤N∏

i=1

||∂if ||1N

L1

for f ∈ C∞0 (Rm) and N ≥ 2, which is elementary, but deep.

Lemma 12. Let µ be a finite measure on RN and assume that there are con-stants ci for i = 1, ..., N such that∣∣∣∣∫

Rm

∂iφdµ

∣∣∣∣ ≤ ci||φ||∞

for all φ ∈ C∞b (Rm), then µ is absolutely continuous with respect to the Lebesguemeasure.

3. EXISTENCE OF DENSITIES 51

Proof. We show the case for N ≥ 2: we shall show that the density of µbelongs to L

NN−1 for N > 1. We take a Dirac sequence ψε for ε > 0 and a sequence

of smooth bump functions 0 ≤ cM ≤ 1 with

cM (x) =

1 for ||x|| ≤M0 for ||x|| ≥M + 1

where we assume that the partial derivatives are bounded uniformly with respectto M . Then the functions cM ·ψε ∗µ belong to C∞0 (RN ). We apply the Gagliardo-Nirenberg inequality and have to estimate additionally

||∂i(cM · ψε ∗ µ)||L1 ≤∫

RN

cM (x) |∂i(ψε ∗ µ)(x)| dx

+∫

RN

|∂icM (x)| (ψε ∗ µ)(x)dx

≤∫

RN

|ψε(x− y)νi(dy)| dx

+∫

RN

|∂icM (x)| (ψε ∗ µ)(x)dx

where νi denotes the signed finite measure on RN induced by φ 7→∫∂iφdµ for

φ ∈ C∞b (RN ). This expression is bounded by a constant independent of M and ε

by Fubini’s theorem. Now the unit ball of LN

N−1 is weakly compact, so we find aweak limit of cM · ψε ∗ µ in L

NN−1 , which is the density of µ, more precisely:∫

RN

g(x)cM (x) · (ψε ∗ µ)(x)dx→∫

RN

g(x)µ(dx)

for g ∈ L∞(RN ) as M →∞ and ε→ 0. Since there exists a weak limit p we obtain∫RN

g(x)µ(dx) =∫

RN

g(x)p(x)dx

which is the desired result. The case N = 1 can be dealt with by similar methods.

Take now a random vector F := (F 1, ..., FN ), which belongs to D1,1loc com-

ponentwise. We associate to F the Malliavin (covariance) matrix γF , which is anon-negative, symmetric random matrix:

γF := (⟨DF i, DF j

⟩H

)1≤i,j≤N .

From regular invertibility of this matrix we shall obtain the basic condition on theexistence of a density:

Theorem 27. Let F be a random vector satisfying

(1) F i ∈ D2,4loc for all i = 1, ..., N .

(2) The matrix γF is invertible almost sure.

Then the law of F is absolutely continuous with respect to Lebesgue measure.


Proof. We shall assume F i ∈ D2,4 for each i = 1, ..., N first. We fix a testfunction φ ∈ C∞0 (RN ), then by Proposition 5 φ(F ) ∈ D1,4, consequently

D(φ(F )) =N∑

i=1

∂φ

∂xi(F )DF i,

⟨D(φ(F )), DF j

⟩H

=N∑

i=1

∂φ

∂xi(F )γij

F

and therefore by invertibility

∂φ

∂xi(F ) =

N∑j=1

⟨D(φ(F )), DF j

⟩H

(γ−1F )ij .

In the sequel we have to apply a localization argument: consider the compact subsetKm ⊂ GL(N) of matrices σ with |σij | ≤ m and |det(σ)| ≥ 1

m for i, j = 1, ...,m.We can define ψm ∈ C∞0 (MN (R)) with ψm ≥ 0, ψm|Km = 1 and ψm|Km+1 = 0,which is easily possible since Km is an exhaustion of GL(m) by compact sets suchthat Km ⊂ (Km+1). Now we can integrate reasonably the above equation

E(ψm(γF )∂φ

∂xi(F )) =

N∑j=1

E(ψm(γF )⟨D(φ(F )), DF j

⟩H

(γ−1F )ij).

Remark that ψm(γF )DF j(γ−1F )ij ∈ dom1,2(δ), since ψm(γF )(γ−1

F )ij ∈ dom1,2(δ) (itequals the inversion rational function applied to γF times a smooth function withcompact support applied to γF , but γF ∈ D2,4) is a bounded random variable and

E((ψm(γF )(γ−1

F )ij)2 ⟨

DF j , DF j⟩

H

)<∞.

Consequently we can apply the duality relationship to arrive at

E(ψm(γF )∂φ

∂xi(F )) = E(φ(F )

N∑j=1

δ(ψm(γF )φ(F )DF i(γ−1

F )ij))

≤ ||φ||∞E

∣∣∣∣∣∣N∑

j=1

δ(ψm(γF )φ(F )DF i(γ−1

F )ij)∣∣∣∣∣∣ .

By Lemma 12 we obtain that the measure F∗(ψm(γF )P ) is absolutely continuouswith respect to the Lebesgue measure on RN . Thus for any A ∈ B(RN ) with zeroLebesgue measure we obtain ∫

F−1(A)

ψm(γF )dP = 0,

but as m → ∞ – via property 2 of the assumptions –∫

F−1(A)dP = 0. Therefore

F∗P λ.In general – for F ∈ D2,4

loc – we calculate for Fn and obtain the result by theproperty that F−1

n (A) → F−1(A).

Next we shall work on the question of regularity properties of the existingdensity, therefore we need a considerable extension of Lemma 12.

4. HORMANDER’S THEOREM IN FINITE DIMENSIONS 53

Lemma 13. Let µ be a finite measure on RN and A ⊂ RN open. Assume thatthere are constants cα for a multiindex α such that∣∣∣∣∫

RN

∂αφdµ

∣∣∣∣ ≤ cα||φ||∞

for all φ ∈ C∞b (RN ) with compact support in A, then the restriction of µ to Ais absolutely continuous with respect to the Lebesgue measure and the density issmooth.

Theorem 28. Let F be a random vector satisfying

(1) F i ∈ D∞ for all i = 1, ..., N .(2) The matrix γF is invertible almost sure and 1

det(γF ) ∈ L∞−0.

Then the law of F is absolutely continuous with respect to Lebesgue measureand the existing density is smooth.

The previous Lemma and Theorem are proved by iterated application of theabove methods.

4. Hormander’s Theorem in finite dimensions

We shall provide a short account on stochastic differential equations designedfor the purposes of these lecturenotes: We shall always assume the following con-ditions on vector fields X : RM × RN → RN :

(1) X are measurable.(2) There is a constant C such that ||X(y, x1)−X(y, x2)|| ≤ C||x1 − x2|| for

all x1, x2 ∈ Rm and all y ∈ RM .(3) The function ||X(y, x)|| is bounded by a constant polynomial in ||y|| for

all x ∈ Rm.

Under these conditions we can prove the following fundamental existence resultwith a Picard iteration scheme:

Theorem 29. Let T > 0 and given a probability space (Ω,F , P ) together with ad-dimensional Brownian motion (Bt)0≤t≤T . Let V, V1, ..., Vd be vector fields satisfy-ing the above conditions and assume that there are a continuous, adapted RM -valuedprocess (Zt)t≥0 with

supt∈[0,T ]

||Zt||p <∞

for all p ≥ 2 and a continuous adapted RN -valuedprocess (αt)t≥0 with

supt∈[0,T ]

||αt||q <∞

for some q ≥ 2, then the stochastic differential equation

Xt = αt +∫ t

0

V (Zs, Xs)ds+d∑

i=1

∫ t

0

Ai(Zs, Xs)dBis

has a unique continuous adapted solution (Xt)0≤t≤T with

supt∈[0,T ]

||Xt||q <∞.


Furthermore the solution can be constructed as Lq-limit of the iteration scheme

Xn+1t = αt +

∫ t

0

V (Zs, Xns )ds+

d∑i=1

∫ t

0

Vi(Zs, Xns )dBi

s

X0t = αt

for 0 ≤ t ≤ T .

This fundamental result leads to the following observations: Given a probabilityspace (Ω,F , P ) together with a d-dimensional Brownian motion (Bt)0≤t≤T in itsnatural filtration, we can ask whether the solution of the equation

dXxt = V (Xt)dt+

d∑i=1

Vi(Xxt )dBi

t(4.1)

Xx0 = x

for x ∈ RN lies in D1,2(Ω; RN ), where we work with the isonormal Gaussian processη : L2([0, T ],Rd) → L2(Ω)

η(h) =d∑

i=1

∫ T

0

hi(s)dBis.

We assume the following conditions on the vector fields V, V1, ..., Vd : RN → RN :(1) The vector fields are smooth.(2) All derivatives of order higher than 1 are bounded.

These conditions are usually refered to as C∞-boundedness conditions. Theyimply that the vector fields satisfy the assumptions of Theorem 29 and in particularXx

t ∈ L∞−0(Ω; RN ) := ∩p≥1Lp(Ω; RN ) for x ∈ RN .

Theorem 30. Let V, V1, ..., Vd : RN → RN be vector fields satisfying C∞-boundedness conditions, then Xx

t ∈ D∞ := ∩p≥1,k≥1Dk,p for 0 ≤ t ≤ T . The firstMalliavin derivative satisfies the following stochastic differential equation

DkrX

xt = V k(Xx

r ) +∫ t

r

dV (Xxs )Dk

rXxs ds+

∫ t

r

dVi(Xxs )Dk

rXxs dB

is

for 0 ≤ r ≤ t ≤ T .

Proof. For the proof we apply two observations. Given u ∈ D1,2(Ω; Rm)progressively measurable, then for t ≥ r

Dr

∫ t

0

usds =∫ t

r

Drusds

by Riemannian approximations and closedness of the operator. Notice furthermorethat Drus = 0 almost surely if r > s. Given u = (u1, ..., ud) ∈ D1,2(Ω; RN ) ⊗ Hwith i = 1, ..., d progressively measurable, then for t ≥ r

Dkr

∫ t

0

d∑i=1

ui(s)dBis =

∫ t

r

d∑i=1

Dkrui(s)dBi

s + uk(r),

again by Riemannian sums and closedness of the Malliavin derivative operator.Going to the Picard approximation scheme we can apply these results to obtain asequence Xn

t ∈ L∞−0 with Xn ∈ D1,p for p ≥ 2 by induction and the chain rule


for n ≥ 0. The derivatives converge to the solution of the stochastic differentialequation, so we conclude by closedness. The solution of this stochastic differentialequation exist due to Theorem 29. For higher derivatives we proceed by induction.

In order to find a good representation of the Malliavin Derivative, we introducefirst variations of the solution of the stochastic differential equation:

dJs→t(x) = dV (Xxt ) · Js→t(x)dt+(4.2)

+d∑

i=1

dVi(Xxt ) · Js→t(x)dBi

t,

Js→s(x) = idN ,(4.3)

for t ≥ s. A similar equation is satisfied by the Malliavin derivative itself. Theequation for the inverse of Js→t(x) looks different, namely

d(Js→t(y, η))−1 = −Js→t(x)−1 · dV (Xxt )dt−(4.4)

−d∑

i=1

Js→t(x)−1 · dVi(Xxt )dBi

t.

The second equation can be solved, hence we know, by Ito calculus, that the matrixJs→t(x) is almost surely invertible, since the solution of the second equation is theinverse. This is due to the well-known equation for matrix valued continuous process

XtYt = X0Y0 +∫ t

0

XsdYs +∫ t

0

dXsYs +12〈X,Y 〉t .

Calculating the semi-martingale decomposition of (J0→t(x))−1J0→t(x)) yields theresult, namely

(J0→t(x))−1J0→t(x) = idN ,

hence the statement on invertibility is justified. Furthermore, we are able to writethe Malliavin derivative,

DisX

xt = J0→t(x)J0→s(x)−1Vi(Xx

s )1[0,t](s).

This is due to the fact that the RN -valued solution process (Yt)r≤t≤T of

Yt = V k(Xxr ) +

∫ t

r

dV (Xxs )Ysds+

∫ t

r

dVi(Xxs )YsdB

is,

is given through

Yt = J0→t(x)J0→r(x)−1V k(Xxr )

for r ≤ t.We give ourselves a scalar product on RN , then we can calculate the covariance

matrix with respect to a orthonormal basis, i.e.

〈γ(Xxt )ξ, ξ〉 :=

d∑i=1

∫ t

0

⟨J0→t(x)J0→s−(y)−1Vi(Y

y,ηs− ), ξ

⟩2ds.


Consequently, the covariance matrix can be calculated via the reduced covari-ance matrix ⟨

Ctξ, ξ⟩

:=d∑

i=1

∫ t

0

⟨J0→s−(y)−1Vi(Y

y,ηs− ), ξ

⟩2ds,

γ(Xxt ) = J0→t(x)CtJ0→t(x)T.

In order to show invertibility of γ(Xxt ) it is hence sufficient to show it for Ct, since

the first variation process J0→t(x) is nicely invertible.The argument for invertibility (which is much simpler) is shown in the next

section (on infinite dimensional systems). We shall focus here on the argument,why in fact 1

det(γ(Xxt ) ∈ ∩p≥1L

p(Ω). The proof is divided in several steps and weshall need Hormander’s assumption for it, i.e. that

(4.5) 〈V1(x), . . . Vd(x), [Vi(x), Vk(x)](i = 0, . . . , d), . . . 〉 = RN

for all x ∈ RN in a uniform way, i.e. there exists a finite number of vector fieldsX1, . . . , XM generated by the above procedure through Lie-bracketing and c > 0such that

(4.6) infξ∈SM−1

M∑k=1

〈Xk(x), ξ〉2 ≥ c

for all x ∈ RN . Here we apply Stratonovich drift vector field, i.e.

V0(x) := V (x)− 12

d∑i=1

DVi(x) · Vi(x).

Remark 12. One can replace the uniform Hormander condition by one, whichonly holds at one point x ∈ RN ,but for the sake of simplicity we shall work with theuniform condition. In the following sketch one has to apply stopping times in thenon-uniform case.

Remark 13. We are working with Ito diffusions, i.e. continuous adapted pro-cesses Xt of the form

Xt = X0 +∫ t

0

v(s)ds+d∑

i=1

∫ t

0

ui(s)dBis,

where we assume all processes in question to be progressively measurable and sat-isfy some integrability assumptions. Notice that this decomposition into a finitevariation process and a martingale is unique. For two Ito processes X and Y thequadratic variation process (〈X,Y 〉t)0≤t≤T is a continuous, adapted process givenby

〈X,Y 〉t =∫ t

0

(d∑

i=1

uXi (s)(uY

i (s))T)ds.

The Stratonovich integral is then defined∫ t

0

Xt dYt :=∫ t

0

XtdYt +12〈X,Y 〉t


We can therefore write by Ito’s formula for Ito diffusions

df(Xt) = (df)(Xt)dXt +12(d2f)(Xt)(dXt)(dXt)

= (df)(Xt) dXt.

Consequently the Stratonovich calculus is of first order, however, we can only in-tegrate continuous semimartingales. Given the solution of equation (4.1), we cantransform (since only nice semi-martingales are integrated) to Stratonovich notationand obtain

dXxt = V0(Xx

t )dt+d∑

i=1

Vi(Xxt ) dBi

t,

with the Stratonovich drift.

We sketch the necessary prerequisites for the following theorem:

Theorem 31. Let (Ω,F , P, (Ft)t≥0) be a filtered probability space and let (Bt)t≥0

be a d-dimensional Brownian motion adapted to the filtration (which is not neces-sarily generated by the Brownian motion). Let V, V1, . . . , Vd, the diffusion vectorfields be C∞-bounded on RM and consider the solution (Xx

t )0≤t≤T of a stochas-tic differential equation (in Stratonovich notation). V0 denotes the Stratonovichcorrected drift term,

dXxt = V0(Xx

t )dt+d∑

i=1

Vi(Xxt ) dBi

t,(4.7)

Xx0 = x.(4.8)

Assume uniform Hormander condition (4.5). Then for any p ≥ 1 we find numbersε0(p) > 0 and an integer K(p) ≥ 1 such that for each 0 < s < T

supξ∈SM−1

P (〈Csξ, ξ〉 < ε) ≤ εp

holds true for 0 ≤ ε ≤ sK(p)ε0(p). The result holds uniformly in x.

The proof of the theorem is a careful re-reading of Norris’ Lemma and theclassical proof of Hormander’s theorem in probability theory (see [27] or [30]). Weshall sketch this path in the sequel (see [30], pp.120–123):

(1) Consider the random quadratic form

〈Csξ, ξ〉 =d∑

i=1

∫ s

0

⟨J0→u(x)−1Vi(Xx

u), ξ⟩2du.

Following the proof presented in [30], we define

Σ′0 := V1, . . . , Vd

Σ′n :=

[Vk, V ], k = 1, . . . , d, V ∈ Σ′n−1; [V0, V ] +

12

d∑i=1

[Vi,[Vi, V ]], V ∈ Σ′n−1

for n ≥ 1. Then we know that there exists j0 such that

infξ∈SM−1

j0∑j=0

∑V ∈Σ′j

〈V (x), ξ〉2 ≥ c


uniformly in x ∈ RM .(2) We define m(j) := 2−4j for 0 ≤ j ≤ j0 and the sets

Ej := ∑

V ∈Σ′j

∫ s

0

⟨J0→u(x)−1V (Xx

u), ξ⟩2du ≤ εm(j).

We consider the decomposition

E0 = 〈Csξ, ξ〉 ≤ ε ⊂ (E0 ∩ Ec1) ∪ (E1 ∩ Ec

2) ∪ · · · ∪ (Ej0−1 ∩ Ecj0) ∪ F,

F = E0 ∩ · · · ∩ Ej0 .

and proceed with

P (F ) ≤ Cεqβ2 ,

for ε ≤ ε1. Furthermore 0 < β < m(j0), any q ≥ 2, a constant C depend-ing on q and the norms of the derivatives of the vector fields V0, . . . , Vd.The number ε1 is determined by the following two (!) equations

(j0 + 1)εm(j0)1 <

cεβ14,

εβ1 < s.

Hence ε1 depends on j0, c, s and the choice of β, via

ε1 < min

(s

1β ,

(c

4(j0 + 1)

) 1m(j0)−β

)(3) We obtain furthermore that

P (Ej ∩ Ecj+1)

≤∑

V ∈Σ′j

P

(∫ s

0


u), ξ⟩2du ≤ εm(j),

d∑k=1

∫ s

0

⟨J0→u(x)−1[Vk, V ](Xx

u), ξ⟩2du+

+∫ s

0

⟨J0→u(x)−1

([V0, V ] +

12

d∑i=1

[Vi, [Vi, V ]]

)(Xx

u), ξ

⟩2

du >εm(j+1)

n(j)

,

where n(j) = #Σ′j . Since we can find the bounded variation and thequadratic variation part of the martingale (


u), ξ⟩)0≤u≤s

in the above expression, we are able to apply Norris’ Lemma (see [30],Lemma 2.3.2). We observe that 8m(j + 1) < m(j), hence we can apply itwith q = m(j)

m(j+1) .(4) We obtain for p ≥ 2 – still by the Norris’ Lemma – the estimate

P (Ej ∩ Ecj+1) ≤ d1

(εm(j+1)

n(j)

)rp

+ d2 exp(−(εm(j+1)

n(j))−ν

)for ε ≤ ε2. Furthermore r, ν > 0 with 18r+9ν < q−8, the numbers d1, d2

depend on the vector fields V0, . . . , Vd, and on p, T . The number ε2 canbe chosen like ε2 = ε3s

k1 , where ε3 does not depend on s anymore.


(5) Putting all together we take the minimum of ε1and ε2 to obtain the desireddependence on s.

Given the previous theorem it is easy to conclude the general statement byfollowing Lemma, which proof can be found in [30].

Lemma 14. Given a random matrix γ ∈ ∩p≥1Lp(Ω) and assume that for p ≥ 1

there is ε0(p) such that

supξ∈SM−1

P (〈Csξ, ξ〉 < ε) ≤ εp

for 0 ≤ ε ≤ ε0(p), then 1det(γ) ∈ ∩p≥1L

p(Ω).

We can also apply directly the nice invertible covariance matrix, which leadsto ”Calculation of Greeks” ideas.

Definition 22. Let (Xxt )t≥0 denote the solution of equation (4.1) and assume

(4.6). Fix t > 0 and x ∈ RN . Fix a direction v ∈ RN . We define a set ofSkorohod-integrable processes

At,x,v = a ∈ dom(δ) such thatd∑

i=1

∫ t

0

J0→s(x)−1Vi(Xxs )ai

sds = v.

Proposition 7. Let (Xxt )t≥0 denote the unique solution of equation (4.1) and

assume d = N . Fix t > 0 and x ∈ RN . Assume furthermore uniform ellipticity,i.e., there is c > 0 such that

infξ∈SM−1

N∑k=1

〈Vk(x), ξ〉2 ≥ c.

Then At,x,v 6= ∅ and there exists a real valued random variable π (which dependslinearly on v) such that for all bounded random variables f we obtain

d

dε

∣∣∣∣ε=0

E(f(Xx+εvt )) = E(f(Xx

t )π).

Proof. Here the proof is particularly simple, since we can take a matrixσ(x) := (V1(x), . . . , VN (x)), which is uniformly invertible with bounded inverse.We define

as := σ(Xxs−)−1 · v

for 0 ≤ s ≤ t and obtain that a ∈ At,x,v. Furthermore – as in [19] (see also theproof of the following theorem) –

π =d∑

i=1

∫ t

0

aisdB

is,

since the Skorohod integrable process a is in fact adapted, left-continuous and henceIto-integrable.

Theorem 32. Let (Xxt )t≥0 denote the unique solution of equation (4.1) and

assume (4.6). Fix t > 0 and x ∈ RN . Fix a direction v ∈ RN . Then At,x,v 6= ∅and there exists a real valued random variable π (which depends linearly on v) suchthat for all bounded random variables f we obtain

d

dε

∣∣∣∣ε=0

E(f(Xx+εvt )) = E(f(Xx

t )π).


We can choose π to be the Skorohod integral of any element a ∈ At,x,v 6= ∅.

Proof. We take f bounded with bounded first derivative, then we obtain

d

dε

∣∣∣∣ε=0

E(f(Xx+εvt )) = E(df(Xx

t )J0→t(x) · v).

If there is a ∈ At,x,v, we obtain

E(df(Xxt )J0→t(x) · v) = E(df(Xx

t )d∑

i=1

∫ t

0

J0→t(x)J0→s(x)−1Vi(Xxs )ai

sds)

= E(d∑

i=1

∫ t

0

df(Xxt )J0→t(x)J0→s(x)−1Vi(Xx

s )aisds)

= E(d∑

i=1

∫ t

0

Disf(Xx

t )aisds)

= E(f(Xxt )δ(a)).

Here we cannot assert that the strategy is Ito-integrable, since it will anticipativein general. In order to see that At,x,v 6= ∅ we construct an element, namely

ais :=

⟨J0→s(x)−1Vi(Xx

s ), (Ct)−1v⟩,

where Ct denotes the reduced covariance matrix. Indeedd∑

i=1

⟨∫ t

0

J0→s(x)−1Vi(Xxs )ai

sds, ξ

⟩

=d∑

i=1

∫ t

0


s ), ξ⟩ ⟨J0→s(x)−1Vi(Xx

s ), (Ct)−1v⟩ds

=⟨ξ, Ct(Ct)−1v

⟩= 〈ξ, v〉

for all ξ ∈ RN , since Ct is a symmetric random operator defined via

⟨ξ, Ctξ

⟩=

d∑i=1

∫ t

0


s ), ξ⟩2ds

for ξ ∈ RN .

For any other derivative with respect to parameters ε, we consider a modifiedset of

Bt,x,v = b ∈ dom(δ) such thatd∑

i=1

∫ t

0

J0→s(x)−1Vi(Xxs )bisds = J0→t(x)

d

dε

∣∣∣∣ε=0

Xx,εt .

Here we are given a parameter-dependent process Xx,εt , where all derivatives with

respect to ε can be calculated nicely. Also in this case we can construct – if thereduced covariance matrix is invertible – an element, namely

bis :=⟨J0→s(x)−1Vi(Xx

s ), (Ct)−1J0→t(x)d

dε

∣∣∣∣ε=0

Xx,εt

⟩.

5. HORMANDER’S THEOREM IN INFINITE DIMENSIONS 61

This is due to the following reasoning,

d∑i=1

⟨∫ t

0

J0→s(x)−1Vi(Xxs )bisds, ξ

⟩

=d∑

i=1

∫ t

0


s ), ξ⟩⟨

J0→s(x)−1Vi(Xxs ), (Ct)−1J0→t(x)

d

dε

∣∣∣∣ε=0

Xx,εt

⟩ds

=⟨ξ, Ct(Ct)−1J0→t(x)

d

dε

∣∣∣∣ε=0

Xx,εt

⟩=⟨ξ, J0→t(x)

d

dε

∣∣∣∣ε=0

Xx,εt

⟩,

due to symmetry of Ct.

5. Hormander’s Theorem in infinite dimensions

Given a separable Hilbert space H and the generator A of a strongly continu-ous group (sic!). We aim to prove a Hormander theorem for stochastic evolutionequations of the daPrato-Zabczyk type (see [9] for all details)


i=1

σi(rt)dBit,(E0)

r0 ∈ H,

under the assumption, that iterative Lie brackets of the Stratonovich drift andthe volatility vector fields span the Hilbert space. We therefore apply methodsfrom Malliavin calculus, which have already been used to solve similar questions infiltering theory (see for instance [31]) or in stochastic differential geometry (see forinstance [1] and [2]).

A particular example, which received some attention recently (see for instance[6] and [16]) is the Heath-Jarrow-Morton equation of interest rate theory (in thesequel abbreviated by HJM),

drt = (d

dxrt + αHJM (rt))dt+

d∑i=1

σi(rt)dBt,

where H is a Hilbert space of real-valued functions on the real line. The HJM driftterm is given by

αHJM (r)(x) :=d∑

i=1

σi(r)(x)∫ x

0

σi(r)(y)dy

for x ≥ 0 and r ∈ H. In order to apply Theorem 33 to the HJM-equation weintroduce the relevant setting in Section 3.

The HJM-equation describes the time-evoultion of forward rates (which containthe full information of a considered bond market) in the martingale measure. It is ofparticular importance in applications to identify relevant, economically reasonablefactors in this evolution. More precisely: how to find a Markov process with valuesin some finite dimensional state space (the space of economically reasonable factors),such that the whole evolution becomes a deterministic function of this Markovprocess? Conditions in order to guarantee this behaviour have been described in[6] and [16]. Economically reasonable factors are the forward rate itself at some


time to maturity x ≥ 0, or averages drawn from it, so-called Yields. If the time-evolution of interest rates cannot be described by finitely many stochastic factors,we can imagine the following generic behaviour, which we formulate in a criterion.

Criterion 1. We denote by (rt(x))t≥0 a forward rate evolution in the Musielaparametrization, i.e. a mild solution of the HJM equation. For x > 0 the associatedYield is denoted by

Yt(x) :=1x

∫ x

0

rt(y)dy,

we define Yt(0) = Rt = rt(0), the short rate process. The evolution is called genericif for each selection of times to maturity 0 ≤ x1 < · · · < xn, the Rn-valued process(Yt(x1), . . . , Yt(xn)) admits a density with respect to the Lebesgue measure.

Remark 14. In financial mathematics generic evolutions do not seem reason-able, since – loosely speaking – the support of the random variable rt for some t > 0becomes too big in the Hilbert space of forward rate curves. In other words, any”shape” of forward rate curves, which we assume from the beginning to model themarket phenomena, is destroyed with positive probability. Hence the very restrictivephenomenon of finite dimensional realizations for the HJM-equation also appearsas the only structure where ”shape” is not destroyed immediately. Hence genericevolutions behave essentially different from affine, finite dimensional realizations,where we can always find tenors x1 < · · · < xn, such that the Yield process doesnot admit a density.

By [16], the existence of finite dimensional realizations is – among technicalassumptions – equivalent to the fact that the stochastic evolution admits locallyinvariant submanifolds (with boundary). This is equivalent to the fact that a certainLie algebra of vector fields DLA is evaluated to a finite dimensional subspace of theHilbert space H at ”some” points r ∈ H, more precisely there is a natural numberM ≥ 1, such that

dimRDLA(r) ≤M <∞in a dom(A∞)-neighborhood.

In Section 2 we prove the Hormander-type result for evolution equations wherethe drift contains a group generator. We then show in Section 3 that for genericvolatility structures at a point r0 ∈ H, the HJM-equation leads to a generic evolu-tion for the initial value r0.

Conceptually a generic evolution is not desirable in interest rate theory, sincewe expect to exhaust all information by a finite number of Yields. Hence the resultTheorem 34 can be interpreted as an additional argument for finite dimensionalrealizations. Notice also that this result is invariant under the important equivalentchanges of measure: if we obtain a generic evolution with respect to one fixedmeasure, then also with respect to all equivalent measures.

In order to set up the methodological background we refer on the one hand tothe finite dimensional literature in Malliavin Calculus such as [30]. On the otherhand we refer to [16] for the analytical framework, in particular for questions ofdifferentiability of functions on infinite dimensional spaces and for the notion ofderivatives of vector fields V : U ⊂ G→ G, when G is some Frechet space.

We shall mainly work on Hilbert spaces: then the derivative DV : U → L(H)is a linear operator to the Banach space of bounded linear operators, where we canspeak about usual properties as differentiability, boundedness, etc.


We consider evolution equations of the type


i=1

σi(rt)dBit,(E0)

r0 ∈ H,

where A : dom(A) ⊂ H → H is the generator of a strongly continuous group (Tt)t≥0

on a separable Hilbert space H. We apply furthermore the following notations,

dom(Ak) := h ∈ H| h ∈ dom(Ak−1) and Ak−1h ∈ dom(A),

||h||2dom(Ak) :=k∑

i=0

||Aih||2,

dom(A∞) = ∩k≥0 dom(Ak).

The maps α, σ1, . . . , σd : H → dom(A∞) are smooth vector fields with theproperty that α, σ1, . . . , σd : dom(Ak) → dom(Ak) are C∞-bounded. As usual (seefor instance [30]) a vector field V is called C∞-bounded if each higher derivativeDlV : dom(Ak) → Ll(dom(Ak)) is a bounded function for l ≥ 1. In this case Vgrows at most linearly on dom(Ak).

Notice that due to the regularity assumptions we can interpret the equation(E0) also on the Hilbert space dom(Ak), with the same regularity conditions onC∞-boundedness,


i=1

σi(rt)dBit,(Ek)

r0 ∈ dom(Ak).

A global, mild, continuous solution, of equation (Ek) with initial value r0 ∈ dom(Ak)is an adaped stochastic process with continuous paths (rt)t≥0 such that

rt = Ttr0 +∫ t

0

Tt−sα(rs)ds+d∑

i=1

∫ t

0

Tt−sσi(rs)dBis

for t ≥ 0, where T is the group generated by A. Clearly every strong, continuoussolution is a mild, continuous solution by variation of constants (see [9]). We shalloften use the vector field µ, referred to as Stratonovich drift,

µ(r) := Ar + α(r)− 12

d∑i=1

Dσi(r) · σi(r)

for r ∈ dom(A). Notice that the Stratonovich drift is only well-defined on a densesubspace dom(Ak+1) of dom(Ak) for k ≥ 0, if we want µ to take values in dom(Ak).Furthermore µ is not even continuous. We nevertheless have the following regulartiyresult:

Proposition 8. Given equation (Ek), for every r0 ∈ dom(Ak) there is aunique, global mild solution with continuous paths denoted by (rt)t≥0. The nat-ural injections dom(Ak) → dom(Ak+1) leave solutions invariant, i.e. a solution ofequation (Ek) with initial value in dom(Ak+1) is a also a solution of the equationwith index k+ 1. More precisely a mild solution with initial value in dom(Ak+1) is


also a mild solution of the equation with index k + 1, and hence a strong solutionof equation (Ek).

A mild solution of equation Ek with initial value r0 ∈ dom(Ak+1) is a strongsolution of equation Ek, hence the solution process is a semi-martingale and theStratonovich decomposition makes sense,

drt = µ(rt)dt+d∑

i=1

σi(rt) dBit.

If we assume that r0 ∈ dom(A∞), then we can construct a solution process(rt)t≥0 with continuous trajectories in dom(A∞), since the Picard-Lindelof approx-imation procedure converges in every Hilbert space dom(Ak), and the topology ofdom(A∞) is the projective limit of the ones on dom(Ak).

For equations of the above type the following regularity assertions hold true forthe first variation process.

Proposition 9. The first variation equations with respect to (Ek) for k ≥ 0are well-defined on dom(Ak)

dJs→t(r0) · h = (A(Js→t(r0) · h) +Dα(rt) · Js→t(r0) · h)dt+

+d∑

i=1

Dσi(rt) · (Js→t(r0) · h)dBit,

Js→s(r0) · h = h,

for h ∈ dom(Ak), r0 ∈ dom(Ak) and k ≥ 0, t ≥ s. The Stratonovich decompositionon dom(Ak)

(4.9) dJs→t(r0) · h = Dµ(rt) · (Js→t(r0) · h)dt+d∑

i=1

Dσi(rt) · (Js→t(r0) · h) dBit,

is only well-defined for h, r0 ∈ dom(Ak+1), since we need to integrate semi-martingales.The Ito equation has unique global mild solutions and Js→t(r0) defines a continuouslinear operator on dom(Ak), which is invertible if r0 ∈ dom(Ak+1), k ≥ 0. Theadjoint of the inverse (Js→t(r0)−1)∗ admits the Stratonovich decomposition(4.10)

d(Js→t(r0)−1)∗·h = −Dµ(rt)∗·((Js→t(r0)−1)∗·h)dt−d∑

i=1

Dσi(rt)∗·(Js→t(r0)−1)∗·hdBit

for h, r0 ∈ dom(Ak+1) and k ≥ 0, t ≥ s ≥ 0. We have furthermore

Js→t(r0) = J0→t(r0)J0→s(r0)−1

P-almost surely for t ≥ s ≥ 0.

Remark 15. We define a Hilbert space Hk([0, T ]) of progressively measurableprocesses (rs)0≤s≤T such that

E( sups∈[0,T ]

||rs||2dom(Ak)) <∞.

Solutions of equations (Ek) can be viewed as mappings r0 7→ (rt)0≤t≤T . ThenJ0→T (r0) ·h is the derivative of this map in the respective locally convex structures.


Remark 16. For the proof of Proposition 9 we need the property that A and−A generate a semi-group, which is equivalent to the assertion that A generates astrongly continuous group (see [9] for further references). Under this assumptionwe can solve the equations for J0→s(r0) ·h in the Hilbert spaces dom(Ak) and obtaininvertibility as asserted. If A does not generate a strongly continuous group, thefirst variations will not be invertible in general.

Proof. Under our assumptions the regularity assertions are clear, also thecalculation of the first variations (see [9] for all details). The only point left is thatwe are allowed to pass to the Stratonovich decomposition, which is correct, since theassertions of Proposition 8 apply and since we integrate semi-martingales by Ito’sformula on Hilbert spaces (see [9]). Fix now r0 ∈ dom(Ak+1) and h ∈ dom(Ak+1),then invertibility follows from the fact that the semi-martingale

(⟨Js→t(r0) · h1, (Js→t(r0)−1)∗ · h2

⟩dom(Ak)

)t≥s≥0

is constant by the respective Stratonovich decomposition, which leads to⟨Js→t(r0)−1 · Js→t(r0) · h1, h2

⟩dom(Ak)

= 〈h1, h2〉dom(Ak)

for h1, h2 ∈ dom(Ak+1). From this we obtain left invertibility by continuity.To prove that the left inverse also is a right inverse we shall apply the fol-

lowing reasoning. Given an ortho-normal basis (gi)i≥1 of dom(Ak) which lies indom(Ak+1), we can easily compute the semi-martingale decomposition of

N∑i=1

⟨Js→t(r0)−1 · h1, gi

⟩dom(Ak)

〈gi, Js→t(r0)∗ · h2〉dom(Ak) =

N∑i=1

⟨h1, (Js→t(r0)−1)∗ · gi

⟩dom(Ak)

〈Js→t(r0) · gi, h2〉dom(Ak) ,

for h1, h2 ∈ dom(Ak+1) and N ≥ 1. Now we apply the Stratonovich decomposition:by adjoining we can free the gi’s and pass to the limit, which yields vanishing finitevariation and martingale part. Hence⟨

Js→t(r0)Js→t(r0)−1 · h1, h2

⟩dom(Ak)

=

limN→∞

N∑i=1

⟨Js→t(r0)−1 · h1, gi

⟩dom(Ak)

〈gi, Js→t(r0)∗ · h2〉dom(Ak)

= 〈h1, h2〉dom(Ak) ,

which is the equation for the right inverse.Finally the process (J0→t(r0)J0→s(r0)−1)t≥s satisfies the correct differential

equation and we obtain by uniqueness the desired assertion on the decompositionof the first variation process Js→t(r0).

Crucial for the further analysis is the notion of the Lie bracket of two vectorfields V1, V2 : dom(A∞) → dom(A∞) (see [16] for the analytical framework). Weneed to leave the category of Hilbert spaces, since the vector field µ is only well-defined on dom(A∞) as a smooth vector field. We define

[V1, V2](r) := DV1(r) · V2(r)−DV2(r) · V1(r)

for r ∈ dom(A∞).


Fix r0 ∈ dom(A∞). We then define the distribution D(r0) ⊂ H, which isgenerated by σ1(r0), . . . , σd(r0) and their iterative Lie brackets with the vectorfields µ, σ1, . . . , σd, evaluated the point r0. Notice that a priori the direction µ(r0)does not appear in the definition of D(r0),

D(r0) = 〈σ1(r0), . . . , σd(r0), [σi, σj ](r0), . . . , [µ, σi](r0), . . . 〉

As in the finite dimensional case, following the original idea of Malliavin [26],the main theorem is proved by calculation of the (reduced) covariance matrix (seealso [30] for a more recent presentation). We need an additional Lemma on Liebrackets of the type [µ, σi] for i = 1, . . . , d.

Lemma 15. Given a vector field V : dom(Ak) → dom(A∞), then there is asmooth extension of the Lie bracket [µ, V ] : dom(Ak+1) → dom(A∞).

Proof. A vector field V : dom(Ak) → dom(A∞) is well-defined and smoothon dom(A∞) ⊂ dom(Ak). There we define the Lie bracket with µ and obtaina well-defined vector field [µ, V ] : dom(A∞) → dom(A∞). Take for a momentµ(r) = Ar + β(r), where β : dom(Ak) → dom(A∞). Then

[µ, V ](r) := AV (r) +Dβ(r) · V (r)−DV (r) ·Ar −DV (r) · β(r).

Since DV (r) : dom(Ak) → dom(A∞), we obtain a smooth extension on dom(Ak+1).For details and further references on the analysis see [16].

Theorem 33. Fix r0 ∈ dom(A∞) and assume that D(r0) is dense in H. Givenk linearly independent functionals ` := (l1, . . . , lk) : H → R, the law of the process(` rt)t≥0 admits a density with respect to Lebesgue measure on Rk for t > 0.

Proof. Take t > 0. We have to form the Malliavin covariance matrix γt, whichis done by well-known formulas on the first variation (see [30]). The covariancematrix can be decomposed into

γt = (` J0→t(r0))Ct (` J0→t(r0))T,

where the random, symmetric Hilbert-Schmidt-operator Ct, the reduced covarianceoperator, is defined via

〈y, Cty〉 =d∑

p=1

∫ t

0

⟨y, J0→s(r0)−1 · σp(rs)

⟩2ds.

We first show that Ct is a positive operator. We denote the kernel of Ct byKt ⊂H and get a decreasing sequence of closed random subspaces of H. V = ∪t>0Kt

is a deterministic subspace by the Blumenthal zero-one law, i.e. there exists a nullset N such that V is deterministic on N c. We shall do the following calculus onN c.

We fix y ∈ V , then we consider the stopping time

θ := infs, qs > 0

with respect to the continuous semi-martingale

qs =d∑

p=1

⟨y, J0→s(r0)−1 · σp(rs)

⟩2,

Then θ > 0 almost surely and qs∧θ = 0 for s ≥ 0.


Now, a continuous L2-semi-martingale with values in R

ss − s0 =d∑

k=1

∫ s

0

αk(u)dBku +

∫ s

0

β(u)du

for s ≥ 0, which vanishes up to the stopping time θ, satisfies – due to the Doob-Meyer decomposition –

αk(s ∧ θ) = 0for k = 1, . . . , d.

We shall apply this consideration for the continuous semi-martingales ms :=⟨y, J0→s(r0)−1 · σp(rs)

⟩on [0, t] for p = 1, . . . , d. Therefore we need to calculate

the Doob-Meyer decomposition of (ms)0≤s≤t. This can be done simply by applyingequation (4.10) for the adjoint of J0→s(r0)−1,

dms =⟨d(J0→s(r0)−1)∗ · y, σp(rs)

⟩= −

⟨Dµ(rs)∗ · (J0→s(r0)−1)∗ · y, σp(rs)

⟩ds−

−d∑

i=1

⟨Dσi(rs)∗ · (J0→s(r0)−1)∗ · y, σp(rs)

⟩ dBi

s+

+⟨(J0→s(r0)−1)∗ · y,Dσp(rs) · µ(rs)

⟩ds+

+d∑

i=1

⟨(J0→s(r0)−1)∗ · y,Dσp(rs) · σi(rs)

⟩ dBi

s

=⟨(J0→s(r0)−1)∗ · y, [σp, µ](rs)

⟩ds+

d∑i=1

⟨(J0→s(r0)−1)∗ · y, [σp, σi](rs)

⟩ dBi

s.

From the Doob-Meyer decomposition this leads to⟨y, J0→s(r0)−1 · [σp, σi](rs)

⟩= 0⟨

y, J0→s(r0)−1 · [σp, µ](rs)⟩

= 0

for i = 1, . . . , d, p = 1, . . . , d and 0 ≤ s ≤ θ. Notice that all the appearing Liebrackets have a smooth extension to some dom(Ak) for k ≥ 0 due to Lemma 15,where we can repeat the argument recursively.

Consequently the above equation leads by iterative application to⟨y, J0→s(r0)−1 · D(rs)

⟩= 0

for s ≤ θ. Evaluation at s = 0 yields y = 0, since D(r0) is dense in H. Hence Ct is apositive definite operator. Therefore we obtain that there is a null set N , such thaton N c the matrix Ct has an empty kernel. Hence the law is absolutely continuouswith respect to Lebesgue measure, since J0→t(r0) is invertible and therefore γt hasempty kernel (Theorem 2.1.2 in [30], p.86).

Example 6. For instance, if we consider the equation

drt = Artdt+d∑

i=1

hidBit

where h1, . . . , hd ∈ dom(A∞), then it is easily seen that D(r0) is dense in H as soonas the linear span of the orbit (Anhi)n≥0,1≤i≤d is dense in H, for all r0 ∈ dom(A∞).As an example, we can consider H = L2(R, λ), where λ denotes the Lebesgue


measure on R, d = 1, A = ddx and h1 = e−

x22 . This result is well-known and can

be obtained by simpler methods.

Example 7. For non-Gaussian random variables the assertion of Theorem 33is already non-trivial. Let σ(r) = φ(r)h be a smooth vector field, φ : H → R a C∞-bounded function, fix r0 ∈ dom(A∞) such that φ(r0) 6= 0. Then we can calculateconditions such that the Lie algebra at r0 is dense in H.

µ(r) = Ar − 12(φ(r)Dφ(r) · h)h

= Ar + ψ(r)h

Dµ(r) · g = Ag + (Dψ(r) · g)h[µ, σ](r) = φ(r)Ah+ φ(r)(Dψ(r) · h)h− (Dφ(r) ·Ar)h− ψ(r)(Dφ(r) · h)h

= φ(r)Ah+ ψ2(r)h,

hence the span of Anh lies in D(r0) (division by φ around r0 is performed). Con-sequently a necessary condition for D(r0) to be dense in H is that the linear spanof the orbit (Anh)n≥0 is dense in H.

We shall describe a framework for the HJM-equation, where Theorem 33 ap-plies. This framework is narrower than the setting given in [13], but it enables usto conclude without worries the desired result.

• H is a separable Hilbert space of continuous functions on the wholereal line containing the constant functions (constant term structures).The point evaluations are continuous with respect to the topology of theHilbert space. Furthermore we assume that the long rate is well-definedand a continuous linear functional l∞(r) := r(∞) for r ∈ H.

• The shift semigroup (Ttr)(x) = r(t+x) is a strongly continuous group onH with generator d

dx .• The map h 7→ S(h) with S(h)(x) := h(x)

∫ x

0h(y)φ(y)dy for x ≥ 0 (if

x < 0 this relation need not hold true) satisfies

||S(h)|| ≤ K||h||2

for all h ∈ H with S(h) ∈ H for some real constant K. There is a closedsubspace H0 ⊂ H such that S(h) ∈ H if and only if h ∈ H0.

Example 8. The first example and seminal treating of consistency problemsin interest rate theory is outlined in [6]. Here the Hilbert space H is a space ofentire functions, where all the requirements above are fulfilled. In particular theshift group is generated by a bounded operator d

dx on this Hilbert space.

Example 9. Hilbert spaces of the above type can be constructed by methodssimilar to [13], pp.75–81, and can be chosen of the type (for the notations see[13]),

Hw := h ∈ H1loc(R)| ||h||2w :=

∫ ∞

−∞|h′(x)|2w(x)dx+ |h(0)|2 <∞.

Notice that in contrary to [13] we need the forward rate curves to be defined on thewhole real line. The forward curve on the negative real line has no direct financialinterpretation.


We need a further requirement for the volatility vector fields in order that thefunction S is well-defined: we define dom(( d

dx )0) = dom( ddx ) ∩H0 and similar for

all further powers.• The vector fields are smooth maps σi : H → dom(( d

dx )∞0 ) for i = 1, . . . , d.• The restriction σi : U ∩ dom(( d

dx )k) → dom(( ddx )k) are C∞-bounded for

i = 1, . . . , d and k ≥ 0.• The HJM drift term is defined to be

∑di=1 S(σi)

l∞

(d∑

i=1

S(σi)

)= 0,

where l∞ denotes the linear functional mapping a term structure to itslong rate r(∞). By [22] the long rate of an arbitrage free term structuremodel is an increasing process, hence this condition means that we assumeit to be constant.

Under these conditions we can prove the following lemma, which guaranteesthat the hypoellipticity result can be applied.

Lemma 16. Let the above conditions be in force. Then the Hilbert space H0 :=ker l∞ of term structures vanishing at ∞ is an invariant subspace of the HJMequation, furthermore

l∞(rt) = r∗(∞)is deterministic for t ≥ 0 for any solution (rt)t≥0 with initial value r∗ of the HJMequation.

Proof. We take a mild solution of the HJM equation with initial value r∗,

rt = Ttr∗ +

∫ t

0

Tt−sαHJM (rs)ds+d∑

i=1

∫ t

0

Tt−sσi(rs)dBs

and apply the linear functionals l∞ to this equation. By continuity we obtain

l∞(rt) = l∞(Ttr∗) +

∫ t

0

l∞(Tt−sαHJM (rs))ds+d∑

i=1

∫ t

0

l∞(Tt−sσi(rs))dBi

s

= l∞(Ttr∗)

since l∞(Tt−sαHJM (r)) = 0 and l∞(Tt−sσi(r)) = 0 for r ∈ U ⊂ H by our assump-

tions.

With respect to this subspace of codimension 1 we can suppose that the con-dition

D(r0) is dense in H0

holds true, since the only deterministic direction of the time evolution, namely l∞,is excluded.

Theorem 34. Take the above conditions and assume that for some r0 ∈ dom((

ddx

)∞)the condition

D(r0) is dense in H0

holds true, then for linearly independent linear functionals l1, . . . , ln : H0 → R therandom variable (l1(rt), . . . , ln(rt)) has a density.


Proof. We have to restrict the reasoning to H0. Take r0 ∈ H and definer∗ = r0 − r0(∞) ∈ H0 (subtracting the constant term structure at level r0(∞)).With the new vector fields

σi(r) := σi(r + r0(∞))

for r ∈ H0 and i = 1, . . . , d. The solution of the equation associated to these vectorfields at initial value r∗ is given through (rt − r0(∞))t≥0, where (rt)t≥0 denotesthe solution of the original equation with initial value r0. Since the Lie algebraiccondition does not change under translation, we can conclude by Theorem 33 thatfor the given linearly independent l1, . . . , ln, the random variable (l1(rt), . . . , ln(rt))has a density with respect to Lebesgue measure for t > 0.

Corollary 2. Assume that

D(r0) is dense in H0,

then the evolution of the term structure of interest rates is generic.

Proof. For x1 < . . . xn the linear functionals Yi(r) :=∫ xi

0r(y)dy for i =

1, . . . , n are linearly independent as linear functionals on the subspace H0.

APPENDIX A

The Wiener Chaos expansion

We introduce Hermite polynomials by the well-known relation

Hn(x) = n!(−1)nex22dn

dxne−

x22

for n ≥ 1 and H0(x) = 1. The generating function F (x, t) = exp(tx− 12 t

2) satisfiesthe following property

exp(tx− 12t2) =

∑n≥0

tn

n!Hn(x)

due to

exp(tx− 12t2) = exp(

x2

2− 1

2(x− t)2)

= exp(x2

2)∑n≥0

tn

n!dn

dtnexp(−1

2(x− t)2)|t=0

= exp(x2

2)∑n≥0

tn

n!(−1)n dn

dxnexp(−1

2x2).

Given a Gaussian random variable (X,Y ) with expecation m = 0 and

Σ =(

1 E(XY )E(XY ) 1

),

then

E(Hn(X)Hm(Y )) = δnmn!E(XY )n(A.1)

E(Hn(X)) = 0

for n ≥ 1, m ≥ 0 This is simply due to the fact that

E(exp(sX − 12s2) exp(tY − 1

2t2)) = exp(stE(XY ))

for all complex numbers s, t by inserting into the characteristic function. Further-more

E(exp(sX − 12s2)) = 1

for s ∈ C, which yields the second assertion.We denote by RN the countable product of real lines, πn : RN → Rn for n ≥

1 denotes the canonical projection on the first n coordinates, pn : RN → R forn ≥ 1 denotes the canonical projection on the n-th coordinate. By B(RN) wedenote the σ-algebra generated by πn, which coincides with the Borel σ-algebra onRN with respect to the product topology. We consider the probability measures

71

72 A. THE WIENER CHAOS EXPANSION

ν⊗n := ⊗ni=1ν on Rn, the standard normal distributions, which define a probability

measure ν⊗∞ on RN on the σ-algebra B(RN).On L2(ν) we introduce the Hermite polynomials by the following construction:

the polynomials R[x] lie dense in this Hilbert space (since exp(tx) for t ∈ R, is adense subset, see Proposition 3 below), where we define the annihilation d and thecreation operator δ via

dφ = φ′

δφ = −φ′ + xφ

for φ ∈ R[x]. We have the fundamental relation

〈dφ, ψ〉 = 〈φ, δψ〉for all φ, ψ ∈ R[x] by partial integration:

〈dφ, ψ〉 =1√2π

∫Rφ′(x)ψ(x) exp(−x

2

2)dx =

= − 1√2π

∫Rφ(x)

d

dx(ψ(x) exp(−x

2

2))dx =

= − 1√2π

∫Rφ(x)(ψ′(x)− xψ(x)) exp(−x

2

2)dx =

= 〈φ, δψ〉 .

This relation is certainly true on all smooth functions φ with dφ ∈ L2(ν) and ψwith δψ ∈ L2(ν).

Hence Hermite polynomials are also defined via

H0(x) = 1

Hn(x) = δHn−1(x) = (δ)n1.

So Hermite polynomials are polynomials of degree n, furthermore by [d, δ] =dδ − δd = idR[x] (Heisenberg’s relation) we get immediately the following theorem(algebra at work: first establishing the algebraic ”hardware” and then calculatingthe ”soft” results):

Theorem 35. We have the following relations for Hermite polynomials:(1) dHn = nHn−1 for n ≥ 1.(2) The set 1√

n!Hnn≥0 is an orthonormal basis of L2(R, ν).

(3) For L = δd = − d2

dx2 + x ddx on the polynomials we obtain LHn = nHn for

n ≥ 0.(4) Hn+1 = xHn − nHn−1.(5) Given a f ∈ L2(R, ν) smooth with dnf ∈ L2(R, ν) for n ≥ 0, then we

obtainf =

∑n≥0

1n!E(dnf)Hn

(6) The generating function is given by

exp(tx− t2

2) =

∑n≥0

tn

n!Hn(x)

with uniform convergence on compact (x, t)-sets.

A. THE WIENER CHAOS EXPANSION 73

Proof. We did not prove much in a functional analytic sense, anyway byalgebraic considerations we are almost ready. For n = 1 the first assertion is truesince H1(x) = x and δx = 1 = H0(x). We go on by induction and assume theassertion to be true for all n < p for p ≥ 2:

dHp = dδHp−1 = δdHp−1 +Hp−1 = δ(p− 1)Hp−2 +Hp−1 = pHp−1

by induction. For the scalar products we observe that for m ≤ n

〈Hm,Hn〉 = 〈Hm, (δ)n1〉 = 〈dnHm, 1〉

which is 0 for m < n by differentiation and n! for m = n, so orthonormality of thesystem is proved. Linear independence is therefore clear and furthermore it is clearthat the Hermite polynomials generate the polynomials, since for every degree thereis exactly one Hermite polynomial. Hermite polynomials are seen to be eigenvectorsof the generator of the Ornstein-Uhlenbeck semi group by

LHn = nδHn−1 = nHn

for n ≥ 1. The fourth relation follows via

Hn+1 = δHn = −dHn + xHn == xHn − nHn−1

Given f ∈ L2(R, ν) we see that

f =∑n≥0

cnHn

in L2 with cn := 1n! 〈f,Hn〉. Remembering the definition of Hermite polynomials

and applying dnf ∈ L2(R, ν) for all n ≥ 0 we obtain

〈f,Hn〉 = 〈dnf, 1〉 = E(dnf)

which yields the result. The generating function ft := exp(tx− t2

2 ) is seen to be inL2(R, ν) for all t with E(ft) = 1 and

dnft = tnft

E(dnft) = E(tnft) = tn

so we get the almost sure expansion, which is uniformly converging on compact(t, x)-subsets.

We introduce a strongly continuous semigroup on L2(ν):

Ttf(x) :=∫ ∞

−∞f(e−tx+

√1− e−2ty)ν(dy)

for f ∈ L2(ν) and t ≥ 0. The integral is well defined since

(x, y) 7→ f(e−tx+√

1− e−2ty)

is an element of L2(ν⊗2) with∫ ∞

−∞

∫ ∞

−∞f(e−tx+

√1− e−2ty)2ν(dy)ν(dx) = ||f ||2


via a rotation. Therefore Ttf is almost surely well defined and

||Ttf ||2 =∫ ∞

−∞(∫ ∞

−∞f(e−tx+

√1− e−2ty)ν(dy))2ν(dx)

≤∫ ∞

−∞

∫ ∞

−∞f(e−tx+

√1− e−2ty)2ν(dy)ν(dx) = ||f ||2.

Therefore we have a family of contractions Tt on L2(ν), i.e. ||Tt|| ≤ 1. It is asemigroup since T0 = id and

Ts(Ttf)(x) =∫ ∞

−∞

∫ ∞

−∞f(e−t(e−sx+

√1− e−2sy) +

√1− e−2tz)ν(dz)ν(dy)

= (Tt+sf)(x)

again via a rotation in (y, z). Finally the semigroup is strongly continuous, sinceTtf → f for t→ 0 if f is bounded, continuous by dominated convergence and Tt isa contraction:

||Ttf − f || ≤ ||Ttf − Ttg||+ ||Ttg − g||+ ||g − f ||

for f ∈ L2(ν) and g bounded continuous. The semigroup is called Ornstein-Uhlenbeck semigroup. The operator −L = −δd on the polynomials is the restrictionof the generator of the Ornstein-Uhlenbeck semigroup:

d

dt|t=0Ttf(x) = xf ′ − f ′′

We can extend the Hermite expansion to L2(ν⊗n) and L2(ν⊗∞) by the followingdefinitions. For n ≥ 1 we define multi-indices a ∈Zn or a ∈Z(∞) respectively, where|a| :=

∑∞i=1 |ai| (notice that there are only finitely many non-zero-entries) and

denote the index set by Λ. Then for x ∈ Rn or x ∈ R∞ we define

Ha(x) :=∏i≥0

Hai(xi)

with xi := pi(x) for i ≥ 1. We define furthermore for a ≥ 0 the factoriala! :=

∏i≥0 ai! These generalized Hermite polynomials form basis in the respective

spaces:

Lemma 17. The countable families 1√a!Haa≥0 are orthonormal basis of the

spaces L2(ν⊗n) or L2(ν⊗∞).

Proof. The proof is given by abstract nonsense with tensor products:

L2(ν⊗n) ' ⊗ni=1L

2(ν)

where the given basis is mapped to the canonical one by the canonical isomorphism.The infinite tensor product is defined as injective limit of the following injectivesystem for m ≥ n:

⊗ni=1L

2(ν) → ⊗mi=1L

2(ν)f1 ⊗ ...⊗ fn 7→ f1 ⊗ ...⊗ fn ⊗ 1⊗ ...⊗ 1

The infinite tensor product is canonically isomorphic to L2(ν⊗∞) and the givenfamily is the canonical basis under the isomorphism.


Example 10. We define the polynomials for n ≥ 0

Hn(λ, x) := λn2Hn(

x√λ

)

for x ∈ R and λ > 0. For λ = 0 we define Hn(0, x) := xn for n ≥ 0. We then havethe formula

exp(tx− t2λ

2) =

∞∑n=0

1n!tnHn(λ, x)

for λ ≥ 0 uniformly on compacts in R. We observe Hn(1, x) = Hn(x) for n ≥ 0and we know by Theorem 35 that

exp(tx− t2

2) =

∞∑n=0

1n!tnHn(x).

Replacing x by x√λ

and t by t√λ yields the result by definition also in the case

λ = 0:

exp(tx− t2λ

2) =

∞∑n=0

1n!tnHn(λ, x).

Definition 23. Let (Ω,F , P ) be a Gaussian space, then the closed hull of theset Hn(η(h)) for h ∈ H with ||h|| = 1 is called the n-th Wiener chaos and denotedby Hn.

Theorem 36. Let (Ω,F , P ) be a Gaussian space, then the following assertionshold:

(1) For n 6= m the Wiener chaos spaces Hn and Hm are orthogonal.(2) The Hilbert space L2(Ω,F , P ) is can be decomposed as orthogonal direct

sum of Hilbert spaces:

L2(Ω,F , P ) =⊕n≥0

Hn.

(3) For n ≥ 0 and a basis hii≥1 of H, the set Φa for a ∈ Λ and |a| = n,where Λ is the usual index set and

Φa :=1√a!

∏i≥1

Hai(η(hi))

is an orthonormal basis of Hn.

Proof. By Example A.1 for n 6= m the relation

E(Hn(η(h1))Hm(η(h2)) = 0

holds with hi ∈ H and ||hi|| = 1. Consequently the Wiener chaos spaces areorthogonal. Polynomials in η(h1), ..., η(hn) for hi ∈ H and ||hi|| = 1 are densein L2(Ω,F , P ). Since any monomial xn can be uniquely written as sum of Her-mitepolynomials Hr(x) with 0 ≤ r ≤ n for n ≥ 0 we conclude that the direct sumof the Wiener chaos spaces is dense and therefore the Hilbert direct sum is thewhole space.

The elements Φa are certainly orthogonal for different degrees |a|, since at byindependence

E(ΦaΦb) =1√a!

1√b!

∏i≥0

E(Hai(η(hi))Hbi

(η(hi))).


If |a| > |b| ≥ 0 we find at least one i0 ≥ 1 such that ai0 6= bi0 , hence by orthogonalityof the Hermite polynomials we obtain orthogonality of Φa. One could argue withthe tensor product and its canonical isomorphisms, too (see Lemma 17).

On Gaussian probability spaces we can define multiple Wiener-Ito integrals:We need some preparations from Hilbert space theory: given a Hilbert space H,then the n-fold tensor product H⊗n for n ≥ 0 is well-defined. The permutationgroup acts bounded linear on the n-fold tensor product by the following action,π ∈ Sn, then

π(e1 ⊗ ...⊗ en) = eπ(1) ⊗ ...⊗ eπ(n)

which extends to the n-fold tensor product by the universal property. f ∈ H⊗n

is called symmetric if π(f) = f for all π ∈ Sn and for an arbitrary g ∈ H⊗n thesymmetrization g is defined via

g :=1n!

∑π∈Sn

π(g)

which is clearly symmetric. The r-fold (right) contractions are defined by thefollowing formula:

(e1⊗ ...⊗em)⊗r (f1⊗ ...⊗fn) := e1⊗ ...⊗em−r⊗f1⊗ ...⊗fn−r ·r∏

i=1

〈em−r+i, fn−r+i〉

The contraction extends to a bilinear mapping H⊗m × H⊗n → H⊗(m+n−2r) for0 ≤ r ≤ n.

Theorem 37. Let η be an isonormal Gaussian process on some (Gaussian)probability space (Ω,F , P ) with (separable) Hilbert space H. Then there is a uniquecontinuous linear mapping In : H⊗n → L2(Ω,F , P ) for each n ≥ 0 with I0(λ) =λ1Ω for λ ∈ R and I1 = η satisfying the following condition:

In+1(f ⊗ g) = In(f)ηg − nIn−1(f ⊗1 g)

for f ∈ H⊗n and g ∈ H, where n ≥ 1. The properties

In(f) = In(f)

In(⊗j≤me⊗nj

j ) =∏j≤m

Hnj(ηej

)

hold for n ≥ 0, f ∈ H⊗n and an orthonormal basis eii≥0 of H with∑

j≤m nj = n.

Proof. We choose a Hilbert basis eii≥1 and show by induction

In(⊗j≤me⊗nj

j ) =∏j≤m

Hnj (ηej )

if Ip are well-defined bounded maps for p < n and the following assertion assertionis true for p < n given some n ≥ 2:

Ip(π[⊗j≤me⊗nj

j ]) =∏j≤m

Hnj(ηej

)

with∑

j≤m nj = p and any permutation π ∈ Sp. For p = 0 and p = 1 the assertionis true by definition:

I0(πλ) = λ1Ω


for π ∈ S0 = e andI1(πei) = η(ei)

for π ∈ S1 = e. In particular the expression is symmetric, such that we obtainfor p < n: Ip(f) = Ip(f) for all f ∈ H⊗p.

Take f = f1⊗ ...⊗ fn, where fi is an element of the Hilbert basis eii≥1, then

In(f) = In−1(f1 ⊗ ...⊗ fn−1)ηfn − (n− 1)In−2( ˜f1 ⊗ ...⊗ fn−1 ⊗1 fn) Induction=

=∏j≤m

Hnj(ηej

)ηfn− kIn−2(⊗j≤me

⊗nj

j ) Induction=

=∏j≤m

Hnj (ηej )ηfn − k∏j≤m

Hnj(ηej )

with ⊗j≤me⊗nj

j a reordering of f1⊗ ...⊗fn−1 and k the number of times fnappearstherein (since there are (n − 2)!k permutations leaving the last element fn fixed).If k ≥ 1, then the nj = nj for all j ≤ m with ej 6= fn and nj − 1 = nj for ej = fn,otherwise the second term disappears. By the recursive relation

Hn+1(x) = xHn(x)− nHn−1(x)

we get the result, since then for n ≥ 0

In(e⊗ni ) = Hn(η(ei)).

So we have proved invariance under permutations and all the desired assertions.Whence In is a well-defined bounded map, since E((In(⊗j≤me

⊗nj

j ))2) = n! with|n| = n (so bounded on a Hilbert basis).

Definition 24. The maps In : H⊗n → L2(Ω,F , P ) are called (multiple)Wiener-Ito integrals.

Proposition 10. We have the following properties for the maps In : H⊗n →L2(Ω,F , P ):

(1) For n ≥ 0 we have In(H⊗n) = Hn.(2) For f ∈ H⊗n and g ∈ H⊗m we have 〈In(f), Im(g)〉 = δnmn!

⟨f , g⟩

forn,m ≥ 0.

Proof. Since the basis elements ⊗j≤me⊗nj

j are mapped to√

n!Φn! the firstassertion is true. We can finally conclude by the relations on Hermite polynomialsthat

E(In(⊗j≤me⊗nj

j )2) = E(∏j≤m

Hnj(ηej

)2)Independence

=

=∏j≤m

E(Hnj(ηej

)2) =

=∏j≤m

(nj !) = (n!)|| ˜⊗j≤me⊗nj

j ||2

where n =∑

j≤m nj . By polarization we get the identity

E(In(f)Im(g)) = δnmn!⟨f , g⟩


and we observeIn(f ⊗ ...⊗ f) = ||f ||nHn(ηf )

for f ∈ H \ 0 with f = f||f || .

As an important remark we mention, that on infinite dimensional spaces Hilbertspaces the mappings In are already characterized by the following condition

In(h1 ⊗ ...⊗ hn) = ηh1 · ... · ηhn

for h1, ..., hn orthogonal, which is - from the algebraic point of view a strangecondition - since the diagonal elements are left away, but they can be approximatedoff-diagonally (there is lot of space in Hilbert space!) – roughly speaking:

There is certainly a bounde linear mapping satisfying these conditions by thepreceding theorem. To see that it is uniquely defined, we take an isomorphism toL2([0, 1], λ) and calculate thereon. The n-fold tensor products are then isomorphicto L2([0, 1]n, λ). We introduce the space En of linear combinations of functions ofthe form

1A1 ⊗ ...⊗ 1An

where the Ai are pairwise disjoint. On these elementary functions the value of Inis well defined and uniquely given. Furthermore we see immediately as above that

E(In(f)2) = n!||f || = n!||f ||

In(f) = In(f)

where we apply the symmetrization on L2([0, 1]n, λ). We remark that En is densein L2([0, 1]n, λ). Since any function therein can be approximated by functions ofthe type f = 1A1 ⊗ ...⊗ 1An

by a monotone class argument with arbitrary Ai ∈ B,we approximate those by elements of En. We choose m ≥ 1 and divide [0, 1] in 2m

intervals of equal length denoted by Bm,j ,

fm := f ·∑

j1,...,jn

⊗k≤n1Bm,jk

where the sum is taken over all pairwise distinct indices j1, ..., jn ∈ 1, ..., 2m. Thenfm ∈ En for almost all m and fm → f almost sure, so we conclude by dominatedconvergence in L2. The profound reason for this construction is that the diagonalhas measure 0 in [0, 1]n.

We show next for elements of En that

In+1(f ⊗ g) = In(f)ηg − nIn−1(f ⊗1 g)

which yields the result. By linearity we can take f = 1A1⊗...⊗1Anwith Ai pairwise

disjoint and g = 1A with A disjoint from ∪j≤nAj or A = An. In the former casewe conclude by definition since f ⊗1 g = 0. In the latter case we obtain

In+1(1A1 ⊗ ...⊗ 1An⊗ 1An

) = In−1(1A1 ⊗ ...⊗ 1An−1)I2(1An⊗ 1An

)

by an off-diagonal approximation as given above.Now we show the relation for n = 2 and a Borel set A. Dividing A into 2m

subsets Bm,j of measure 12m we get

(η1A)2 =

∑i

(η1Bm,i)2 +

∑i 6=j

(η1Bm,i)(η1Bm,j

) → λ(A) + I2(1A ⊗ 1A)


in L2, which yields I2(1A⊗1A) = (η1A)2−||1A||. This is the desired relation which

defines In uniquely.

Theorem 38. Let (Ω,F , P ) be a Gaussian space, then the following assertionshold:

(1) On the symmetric tensor product, i.e. the closed subspace of symmetrictensors, the multiple Wiener-Ito integrals are injective.

(2) For F ∈ L2(Ω,F , P ) there unique symmetric tensors fn ∈ H⊗n such that

F =∞∑

n=0

In(fn)

This decomposition is called the Wiener chaos expansion.

Proof. The proof is immediate by the contents of this section.

In this last section we shall identify the (multiple) Wiener-Ito integrals asstochastic integrals with respect to Brownian motion and develop some notionsunder the assumption that the Hilbert space H has a concrete representation asL2(T,B, µ). We shall assume that µ is a σ-finite measure without atoms on (T,B).We denote the isomormal Gaussian process η : H → L2(Ω,F , P ) by W in thissection.

We introduce the following notation: H = L2(T,B, µ), then canonically H⊗n =L2(Tn,B⊗n, µ⊗n). An element f ∈ L2(Tn,B⊗n, µ⊗n) is called symmetric if

f(t1, ..., tn) = f(tπ(1), ..., tπ(n))

for π ∈ Sn and n ≥ 1. The symmetrization is given through

f(t1, ..., tn) =1n!

∑π∈Sn

f(tπ(1), ..., tπ(n))

by canonical isomorphisms on tensor products. We introduce the notation

In(f) =∫

T n

f(t1, ..., tn)W (dt1)...W (dtn).

If f is symmetric and T = R≥0 we can additionally write

In(f) = n!∫ ∞

0

∫ tn

0

...

∫ t2

0

fn(t1, ..., tn)W (dt1)...W (dtn).

If W (h) :=∫∞0h(s)dBs with respect to a 1-dimensional Brownian motion on

(Ω,F , P ), we can identify these formal notations as iterated Ito integrals. Weintroduce ∆n = (t1, ..., tn) ∈ Rn

≥0 with 0 ≤ t1 ≤ ... ≤ tn.

Theorem 39. Let (Ω,F , P ) be a probability space with 1-dimesional Brownianmotion such that W (h) =

∫∞0h(s)dBs is an isonormal Gaussian process for h ∈

L2(R≥0) = H. For f ∈ H⊗n symmetric the iterated integrals

In(f) = n!∫ ∞

0

∫ tn

0

...

∫ t2

0

fn(t1, ..., tn)dBt1 ...dBtk−1

as iterated stochastic integral.


Proof. We shall proceed by induction: The hypothesis shall be that for k ≥ 1the iterated integral

V ktk+1,...,tn

:=∫ tk

0

...

∫ t2

0

f(t1, ..., tn)dBt1 ...dBtk

exists for almost all tk+1, ..., tn and that V k has a measurable version supported by∆n−k that is progressive as a process in tk+1 with parameters tk+2, ..., tn. Further-more

E((V ktk+1,...,tn

)2) =∫ tk

0

...

∫ t2

0

f(t1, ..., tn)2dt1...dtk

We start with V 0 = f and find the stated properties, since there is a measurablesection by Fubini’s theorem. Assuming the assertions for k− 1 ≥ 0, then we defineby the hypothesis

Xkt,tk+1,...,tn

:=∫ t

0

V ktk,...,tn

dBtk

for t ≥ 0, which has an almost sure continuous, progressive version in t for fixedtk+1, ..., tn and measurable in all variables by dependence on parameters. We takesuch a version for each tuple tk+1, ..., tn and define

V ktk+1,...,tn

:= Xktk+1,tk+1,...,tn

which is progressive as a process in tk+1 with parameters tk+2, ..., tn and measurablein all variables. The support condition and the Ito isomorphisms is maintained,too. So the existence of the iterated integrals is established, equality follows by anmonotone class argument: Given any rectangular box in ∆n we get the equalityimmediately, then we can conclude by uniqueness of the Wiener chaos maps In.

Example 11. By the properties of the chaos maps In we know that for h ∈H \ 0

In(1

||h||nh⊗n) = In((

h

||h||)⊗n)

= Hn(η(h

||h||))

In(h⊗n) = ||h||nHn(η(h

||h||)).

We apply this formula in the case H = L2(R≥0) and (Ω,F , P ) a probability spacewith one dimensional Brownian motion (Bt)t≥0 such that F is generated by theBrownian motion. η(H) :=

∫∞0h(s)dBs. By the above formula we obtain

In(1⊗n[0,t]) = Hn(t, Bt)

since h = 1[0,t] has norm ||h|| =√t. We obtain the following nice assertions:

(1) (Hn(s,Bs))s≥0 is an L2-martingale with respect to the natural filtration(Fs)s≥0 associated to Brownian motion.

(2) The integral representation is given by∫ t

0

Hn−1(s,Bs)dBs = Hn(t, Bt)

for n ≥ 1 and t ≥ 0.


This can be proved easily be iterated Wiener-Ito integrals:

In(1⊗n[0,t]) = n!

∫ ∞

0

∫ tn

0

...

∫ t2

0

1⊗n[0,t](t1, ..., tn)dBt1 ...dBtn

and therefore for t ≥ s

E(Hn(t, Bt)|Fs) = n!∫ s

0

∫ tn

0

...

∫ t2

0

1⊗n[0,t](t1, ..., tn)dBt1 ...dBtn

= n!∫ ∞

0

∫ tn

0

...

∫ t2

0

1[0,s](tn)1⊗n[0,t](t1, ..., tn)dBt1 ...dBtn

= n!∫ ∞

0

∫ tn

0

...

∫ t2

0

1⊗n[0,s](t1, ..., tn)dBt1 ...dBtn

= Hn(s,Bs).

The integral representation follows the same footsteps∫ t

0

Hn−1(s,Bs)dBs =

=∫ ∞

0

1[0,t](s)(n− 1)!∫ ∞

0

∫ tn−1

0

...

∫ t2

0

1⊗(n−1)[0,s] (t1, ..., tn−1)dBt1 ...dBtn−1dBs

= (n− 1)!∫ ∞

0

1[0,t](s)∫ s

0

∫ tn−1

0

...

∫ t2

0

1⊗(n−1)[0,t] (t1, ..., tn−1)dBt1 ...dBtn−1dBs

= (n− 1)!∫ ∞

0

∫ s

0

∫ tn−1

0

...

∫ t2

0

1⊗n[0,t](t1, ..., tn−1, s)dBt1 ...dBtn−1dBs

=1nHn(t, Bt),

since the value of the integral does not depend on the name of the integration vari-ables.

APPENDIX B

Cameron-Martin Space and Girsanov’s Theorem

We provide two proofs of the Cameron-Martin-Girsanov Theorem. The firstone - given in Malliavin’s numerical model - explains the simplicity of the assertionand the second one - by martingale methods - proves the assertion in full generality:We are asking whether translation of probability measures on Gaussian spaces areabsolutely continuous with respect to the original measure.

Theorem 40 (Cameron-Martin). We denote the translation by α ∈ R∞ ofmeasurable functions on the (locally convex) vector space R∞ by τα, i.e.

(ταf)(x) = f(x+ α)

for x ∈ R∞. If α ∈ l2, then there exists kα ∈ L∞−0(R∞) := ∩p<∞Lp(R∞) such

that

(B.1)∫

(ταf)(x)γ∞(dx) =∫f(x)kα(x)γ∞(dx)

for all f ∈ L1+0(R∞) := ∪p>1Lp(R∞). Furthermore

kα(x) = exp(∑k≥1

αkxk −12||α||22)

||kα||p = exp(p− 1

2||α||22)

d

dε|ε=0kεα(x) =

∑k≥1

αkxk

for α ∈ l2 and x ∈ R∞ and p ≥ 1. If α /∈ l2 then γ∞(. − α) is not absolutelycontinuous with respect to γ∞.

Proof. We define the following (discrete time) martingale for α ∈ R∞:

Mαn (x) := exp(

n∑k=1

αkxk −12

n∑k=1

α2k)

for x ∈ R∞ and n ≥ 0. Notice that the projections pk : R∞ → R, x 7→ xk for k ≥ 1form a sequence of independent Gaussian random variables N(0, 1) and therefore

83

84 B. CAMERON-MARTIN SPACE AND GIRSANOV’S THEOREM

via the formulas for characteristic functions of Gaussian random variables

E((Mλαn )) = E(exp(λ

n∑k=1

αkxk −λ2

2

n∑k=1

α2k)) = 1

E(exp(pn∑

k=1

αkxk)) = exp(p2

2

n∑k=1

α2k)

E(Mαn (x)|Fm) = E(exp(λ

n∑k=m+1

αkxk −λ2

2

n∑k=m+1

α2k))Mα

m = Mαm

for all α ∈ R∞, n ≥ m ≥ 0 and p ≥ 1 by indepedence of the random variable pk ofFm = σ(pl for l ≤ m) for k > m. The martingale (Mα

n ) is a uniformly integrableL1-martingale if and only if α ∈ l2. This can be seen by evaluating the followingintegral for t ≥ 0:

∫R|1− exp(tx− t2

2)|γ(dx) =

∫ ∞

0

(exp(tx− t2

2)− 1)γ(dx)+

+∫ 0

−∞(1− exp(tx− t2

2))γ(dx)

= (∫ ∞

−t

γ(dx)−∫ −t

−∞γ(dx))

=∫ t

−t

γ(dx),

which converges to 0 if and only if t → 0. Uniformly integrable L1-martingalesconverge in L1 to some limit, so (Mα

n ) uniformly integrable if and only if

E(|Mαn −Mα

m|) = E(Mαm|1− exp(λ

n∑k=m+1

αkxk −λ2

2

n∑k=m+1

α2k))|)

= E(|1− exp(n∑

k=m+1

αkxk −12

n∑k=m+1

α2k))|) → 0

as n ≥ m → ∞ by independence. The Gaussian integral can be evaluated by theabove formula – after a rotation mapping (am+1, ..., αn) 7→ (

√∑nk=m+1 α

2k, ..., 0) –

and we obtain

E(|Mαn −Mα

m|) =∫ √∑n

k=m+1 α2k

−√∑n

k=m+1 α2k

γ(dx)

for n ≥ m, which converges to 0 if and only if α ∈ l2.The bounded, measurable functions on R∞, which depend only on finitely many

variables, form a dense subset of Lp for 1 ≤ p <∞, given f : R∞ → R which factors

B. CAMERON-MARTIN SPACE AND GIRSANOV’S THEOREM 85

to g : Rn → R, i.e. g pn = f for some n ≥ 1, then∫f(x+ α)γ∞(dx) =

∫Rn

g(x+ pn(α))γn(dx)

=∫

Rn

g(x)Mpn(α)n (x)γn(dx)

=∫f(x)Mα

n (x)γ∞(dx)

=∫f(x)kα(x)γ∞(dx),

consequently we the desired equation is valid with kα = limn→∞Mαn , where the

limit is an almost sure limit by martingale convergence, for all bounded measurablefunctions, which depend on finitely many variables. By density in Lp(R∞) theequation holds for all f ∈ L1+0(R∞).

If γ∞(.− α) were absolutely continuous with respect to γ∞, then there wouldexist kα ∈ L1(R∞) such that for all bounded, measurable functions equation B.1holds. Then E(kα|Fm) = Mα

m and therefore the martigale (Mαn ) is uniformly

integrable, which is the case if and only if α ∈ l2, which was the assertion.

Given a Gaussian probability space (Ω,F , P ) with 1-dimensional Brownian mo-tion (Bt)t≥0 with natural filtration (Ft)t≥0 and the associated isonormal Gaussianprocess η : L2(R≥0) → L2(Ω,F , P ), h 7→

∫∞0h(s)dBs, we can draw the following

conclusions.

Theorem 41. Given a progressively measurable process (ut)t≥0 ∈ L(B) on(Ω,F , P ), i.e.

∫ t

0u2

sds < ∞ almost surely for t ≥ 0. For all T ≥ 0 there existsa probability measure QT on FT , absolutely continuous with respect to P |FT

, suchthat the continuous process (Bt −

∫ t

0usds)0≤t≤T has the law of a Brownian motion

under QT on FT , if and only if

E(exp(∫ T

0

usdBs −12

∫ T

0

u2sds)) = 1.

In this case E( dQT

dP |FT|FT ) = exp(

∫ T

0usdBs − 1

2

∫ T

0u2

sds), where Ft denotes the

natural filtration associated to (Bt −∫ t

0usds)0≤t≤T .

Proof. Assume first the integrability condition, then we can consider the in-creasing family of stopping times τk := inft,

∫ t

0u2

sds ≥ k for k ≥ 0 and thecontinuous, positive martingales ξτk

t for k ≥ 0 and t ≥ 0, where

ξt := exp(∫ t

0

usdBs −12

∫ t

0

u2sds).

Hence (ξt) is a continuous local martingale and a martingale since E(ξT ) = 1. Weshall need the following simple technical lemma:

Lemma 18. Given a probability space (Ω,F , P ), η ∈ L1(Ω,F , P ) with η ≥ 0,E(η) = 1 and a sub-σ-algebra G ⊂ F , the probability measure Q with dQ

dP = η iswell defined and

EQ(X|G) =E(Xη|G)E(η|G)

for X ∈ L1(Ω,F , Q) almost sure with respect to Q.


Proof. We can calculate directly by definition of conditional expectations∫EQ(X|G)Y dQ =

∫EQ(X|G)Y ηdP

=∫EQ(X|G)E(η|G)Y dP∫

EQ(X|G)Y dQ =∫XY ηdP

=∫E(Xη|G)Y dP

for Y bounded measurable.

We define a probability measure Q with dQdP = ξT . Consequently by Ito’s

formula dξr = urξrdBr and

dXr = iλXrdBr − iλXrurdr −λ2

2Xrdr

for Xr = exp(iλ(Br −Bs −∫ r

suvdv)) for s ≤ r ≤ t we obtain

d(Xrξr) = Xrdξr + ξrdXr + dXrdξr

= Xrdξr + ξrdXr + iλXrurξrdr

= XrurξrdBr + ξriλXrdBr −λ2

2Xrξrdr,

which yields by conditional Fubini’s theorem

EQ(exp(iλ(Bt −Bs −∫ t

s

urdr))|Fs) =E(exp(iλ(Bt −Bs −

∫ t

surdr))ξT |Fs)

ξs

=E(exp(iλ(Bt −Bs −

∫ t

surdr))ξt|Fs)

ξs

=E(ξs −

∫ t

sλ2

2 Xrξrdr|Fs)ξs

= 1− λ2

2

∫ t

s

E(Xrξr|Fs)ξs

dr

= 1− λ2

2

∫ t

s

EQ(Xr|Fs)dr

and therefore EQ(exp(iλ(Bt −Bs −∫ t

surdr))|Fs) = exp(−λ2

2 (t− s)) for t ≥ s. Byindependence of the increments of Brownian motion we obtain the result.

Assume now that there exists an absolutely continuous probability measure Qsuch that the law of the continuous process Bt := Bt −

∫ t

0u(s)ds is the law of a

Brownian motion for 0 ≤ t ≤ T . Assume that dQT

dP = ψ ∈ L1(Ω,FT , P ). By theabove arguments we know already that

E(f(Bτkt1 , ..., B

τktn

)ξτk

T ) = E(f(Bt1 , ..., Btn))

for bounded measurable f : Rn → R and k ≥ 0. This is true due to the fact that theexponentials exp(η(h)) for h ∈ L2([0, T ]) are dense in L2(Ω,FT , P |FT

). This holdsalso in L2(Ω,FT , QT ) with the shifted Brownian motion, so the identity extends.

B. CAMERON-MARTIN SPACE AND GIRSANOV’S THEOREM 87

On the other hand ξτk

T = E(η|Fτk

T ) for k ≥ 0 by assumption, where Ft denotes thenatural filtration associated to Bt. Letting k →∞ we obtain E(ξT ) = 1.

In the case that ut = h(t) for 0 ≤ t ≤ T with h ∈ L2([0, T ]) the Cameron-MartinTheorem implies the Girsanov Theorem by the following construction: Given a basiseii≥1 of L2([0, T ]), the map

L2(R∞) → L2(Ω)

f 7→ f(η(e1), η(e2), ...)

is an isomorphism. The translation kαf is transformed to

(khF ) = f(η(e1) + 〈e1, h〉 , η(e2) + 〈e2, h〉 , ...)

with F = f(η(e1), η(e2), ...) and h =∑

i≥1 αiei. Notice that the isomorphismdepends on the basis eii≥1, a sort of ”coordinates” for the representation ofrandom variables. The translation kh does not depend on the basis eii≥1, butonly on η. The random variable Bt −

∫ t

0h(s)ds is transformed via kh to Bt, since⟨

1[0,t], h⟩

=∫ t

0h(s)ds, therefore

E(khf(Bt1 −∫ t1

0

h(s)ds, ..., Btn−∫ tn

0

h(s)ds))

= E(f(Bt1 −∫ t1

0

h(s)ds, ..., Btn−∫ tn

0

h(s)ds)ξT )

= E(f(Bt1 , ..., Btn)),

for ti ≤ T , which yields the assertion on the law of the process Bt −∫ t

0h(s)ds

for 0 ≤ t ≤ T . In the case of deterministic strategies in L2([0, T ]) we obtainfurthermore Ft = Ft for t ≥ 0 and therefore uniqueness of Q.

If we interpret this result in the canonical setting Ω = C([0, T ]) we obtain thefollowing picture

H1([0, T ]) → C([0, T ]) → L2([0, T ])

where H1([0, T ]) denotes the first Sobolev space or the Cameron-Martin space. Thecontinuous process (evt)0≤t≤T is a Brownian motion under the Wiener measure Pon C([0, T ]). The Cameron-Martin Theorem tells that

evt(ω)−∫ t

0

h(s)ds = ω(t)−∫ t

0

h(s)ds

= evt(ω −∫ .

0

h(s)ds)

is a Brownian motion under the measure Q with

dQ

dP= exp(

∫ T

0

h(s)dWt −12

∫ T

0

h(s)2ds).

However Q is the translation of P by h, so we obtain the P can be translated byh ∈ C([0, T ]) if and only if h ∈ H1([0, T ]) and

dP (.− h)dP

= exp(∫ T

0

h(s)dWt −12

∫ T

0

h(s)2ds).


Notice that only in the case that h ∈ H1([0, T ]) the translation of a random variableF ∈ L2(Ω) is well defined, since otherwise equivalent measurable functions are nomore equivalent after translation, since null sets are no more preserved. This is thedeep reason for the fact that we defined Malliavin derivatives only with respect tothe directions in the Cameron-Martin space.

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