sode-lecture Stochastic Diﬀerential Equations

8/13/2019 sode-lecture Stochastic Differential Equations

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Contents

1 Stochastic integration 3

1.1 White Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 The Ito Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2.1 Construction inL2 . . . . . . . . . . . . . . . . . . . . . . . . 61.2.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.2.3 Doobs Martingale Inequality . . . . . . . . . . . . . . . . . . 101.2.4 Extension of the Ito integral . . . . . . . . . . . . . . . . . . . 12

1.2.5 The Fisk-Stratonovich integral . . . . . . . . . . . . . . . . . . 131.2.6 Multidimensional Case . . . . . . . . . . . . . . . . . . . . . . 141.2.7 Itos formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2 Strong solutions of SDEs 19

2.1 The strong solution concept . . . . . . . . . . . . . . . . . . . . . . . 19

2.2 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.3 Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.4 Explicit solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.4.1 Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . 262.4.2 Transformation methods . . . . . . . . . . . . . . . . . . . . . 27

3 Weak solutions of SDEs 29

3.1 The weak solution concept . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2 The two concepts of uniqueness . . . . . . . . . . . . . . . . . . . . . 313.3 Existence via Girsanovs theorem . . . . . . . . . . . . . . . . . . . . 32

3.4 Applications in finance and statistics . . . . . . . . . . . . . . . . . . 36

4 The Markov properties 37

4.1 General facts about Markov processes . . . . . . . . . . . . . . . . . . 37

4.2 The martingale problem . . . . . . . . . . . . . . . . . . . . . . . . . 384.3 The strong Markov property . . . . . . . . . . . . . . . . . . . . . . . 404.4 The infinitesimal generator . . . . . . . . . . . . . . . . . . . . . . . . 42

4.5 The Kolmogorov equations . . . . . . . . . . . . . . . . . . . . . . . . 454.6 The Feynman-Kac formula . . . . . . . . . . . . . . . . . . . . . . . . 47


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Table of contents

5 Stochastic control: an outlook 48

Bibliography 53


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Notation

The notation follows the usual conventions, nevertheless the general mathematicalsymbols that will be used are gathered in the first table. The notation of the differentfunction spaces is presented in the second table. The last table shows some regularlyused own notation.

General symbols

A:= B Ais defined by B[a, b], (a, b) closed, open interval froma to bN,N0,Z {1, 2, . . .},{0, 1, . . .},{0, +1, 1, +2, 2, . . .}R,R+,R,C (, ), [0, ), (, 0], complex numbersRe(z), Im(z),z real part, imaginary part, complex conjugate ofz Cx largest integer smaller or equal tox Rx smallest integer larger or equal tox Ra

b, a

b maximum, minimum ofa and b

|x| modulus ofxRor Euclidean norm ofxRdAB Ais contained in B or A = Bspan(v , w , . . .) the subspace spanned byv , w, . . .U+ V, U V the sum, the direct sum (U V ={0}) ofU and Vdim V, codim V linear dimension, codimension ofVran T, ker T range and kernel of the operatorTEd identity matrix inRdd

det(M) determinant of MT,TXY operator norm ofT :XYf(),g(1, 2) the functions xf(x), (x1, x2)g(x1, x2)supp(f) support of the function ff|S function frestricted to the set Sf, f, f(m) first, second,m-fold (weak) derivative offf(a+) derivative of f ata to the right1S indicator function of the setS

f, F(f) f() = F(f)() =R

f(t)eitdt or estimator f offa, F(a),aM(I) a() = F(a)() =

Ieitda(t)

log natural logarithmcos, sin, cosh, sinh (hyperbolic) trigonometric functions


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2 Notation

P,E, Var, Cov probability, expected value, variance and covarianceL(X),X P the law ofX, L(X) = PXn

P

=X Xn convergesP-stochastically to XXn

L=X Xn converges in law to X

N(, 2) normal distribution with mean and variance2

(Zi, iI) -algebra generated by (Zi)iIx Dirac measure atxA B A= O(B), i.e. c >0 p: A(p)cB(p) (p parameter)A B B AA B A B and B A

Function spaces and norms

Lp(I,Rd) p-integrable functionsf :I R (I|f|p


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Chapter 1

Stochastic integration

1.1 White Noise

Many processes in nature involve random fluctuations which we have to account forin our models. In principle, everything can be random and the probabilistic structureof these random influences can be arbitrarily complicated. As it turns out, the socalled white noise plays an outstanding role.

Engineers want the white noise process ( W(t), tR) to have the following prop-erties:

The random variables{ W(t) | tR} are independent.

Wis stationary, that is the distribution of ( W(t + t1), W(t + t2), . . . , W(t + tn))does not depend ont.

The expectationE[ W(t)] is zero.Hence, this process is supposed to model independent and identically distributedshocks with zero mean. Unfortunately, mathematicians can prove that such a real-valued stochastic process cannot have measurable trajectories t W(t) except forthe trivial process W(t) = 0.

1.1.1 Problem. If (t, ) W(t, ) is jointly measurable withE[ W(t)2]


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4 Chapter 1. Stochastic integration

The way out of this dilemma is found by looking at the corresponding integratedequation:

x(t) =x(0) + t0

x(s) ds + t0

W(s) ds, t0.

What properties should we thus require for the integral process W(t) :=t0 W(s)ds,

t0? A straight-forward deduction (from wrong premises...) yields

W(0) = 0.

The increments (W(t1)W(t2), W(t3)W(t4), . . . , W (tn1)W(tn)) are in-dependent fort1t2 tn.

The increments are stationary, that is W(t1+ t)

W(t2+ t)

L=W(t1)

W(t2)

holds for all t0.

The expectationE[W(t)] is zero.

The trajectories tW(t) are continuous.

The last point is due to the fact that integrals over measurable (and integrable) func-tions are always continuous. It is highly nontrivial to show that up to indistinguisha-bility and up to the norming Var[W(1)] = 1 the only stochastic process fulfillingthese properties is Brownian motion (also known as Wiener process) (ksendal 1998).Recall that Brownian motion is almost surely nowhere differentiable!

Rephrasing the stochastic differential equation, we now look for a stochasticprocess (X(t), t0) satisfying

X(t) =X(0) +

t0

X(s)ds + W(t), t0, (1.1.1)

where (W(t), t0) is a standard Brownian motion. The precise formulation involvingfiltrations will be given later, here we shall focus on finding processes Xsolving (1.1.1).

The so-called variation of constants approach in ODEs would suggest the solution

X(t) =X(0)et

+ t0 e(ts) W(s) ds, (1.1.2)which we give a sense (in fact, that was Wieners idea) by partial integration:

X(t) =X(0)et + W(t) +

t0

e(ts)W(s) ds. (1.1.3)

This makes perfect sense now since Brownian motion is (almost surely) continuousand we could even take the Riemann integral. The verification that (1.1.3) defines a


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1.1. White Noise 5

solution is straight forward:

t0 X(s) ds= X(0) t

0 es

ds + t0 W(s) ds + 2 t

0 s

0 e(su)

W(u) duds

=X(0)(et 1) + t0

W(s) ds + 2 t0

W(u)

tu

e(su) dsdu

=X(0)(et 1) + t0

W(u)e(tu) du

=X(t) X(0) W(t).Note that the initial value X(0) can be chosen arbitrarily. The expectation

(t) := E[X(t)] = E[X(0)]et exists if X(0) is integrable. Surprisingly this expec-tation function satisfies the deterministic linear equation, hence it converges to zero

for 0. How about the variation around this mean value?Let us suppose that X(0) is deterministic, = 0 and consider the variance function

v(t) := Var[X(t)] = E

W(t) +

t0

e(ts)W(s) ds2

= E[W(t)2] + 2 t0

e(ts)E[W(t)W(s)] ds + t0

t0

2e(2tus)E[W(s)W(u)] duds

=t + 2

t0

e(ts)s ds + 2

t0

ts

2e(2tus)sduds

=t + t

0 2e(ts)s + 2(e2(ts) e(ts))s ds= 12

e2t 1.

This shows that for < 0 the variance converges to 12|| indicating a stationarybehaviour, which will be made precise in the sequel. On the other hand, for >0 wefind that the standard deviation

v(t) grows with the same order as(t) fort

which lets us expect a very erratic behaviour.In anticipation of the Ito calculus, the preceding calculation can be simplified by

regarding (1.1.2) directly. The second moment oft0

e(ts) dW(s) is immediately seen

to be

t

0e2(ts) ds, the above value.

1.1.2 Problem. Justify the name white noise by calculating the expectation andthe variance of the Fourier coefficients of W on [0, 1] by formal partial integration,i.e. using formally

ak =

10

W(t)

2 sin(2kt) dt= 10

W(t)2k

2 cos(2kt) dt

and the analogon for the cosine coefficients. Conclude that the coefficients are i.i.d.standard normal, hence the intensity of each frequency component is equally strong(white).


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1.2 The Ito Integral

1.2.1 Construction in L2

We shall only need the Ito integral with respect to Brownian motion, so the generalsemimartingale theory will be left out. From now on we shall always be working on acomplete probability space (,F,P) where a filtration (Ft)t0, that is a nested familyof-fields Fs Ft Fforst, is defined that satisfies the usual conditions:

Fs=t>sFt for alls0 (right-continuity);

all A FwithP(A) = 0 are contained in F0.A family (X(t), t0) ofRd-valued random variables on our probability space is calleda stochastic process and this process is (Ft)-adapted if all X(t) are Ft-measurable.

Denoting the Borel -field on [0, ) by B, this process X is measurable if (t, )X(t, ) is a BF-measurable mapping. We say that (X(t), t0) is continuous if thetrajectoriestX(t, ) are continuous for all . One can show that a process ismeasurable if it is (right-)continuous (Karatzas and Shreve 1991, Thm. 1.14).

1.2.1 Definition. A (standard one-dimensional) Brownian motion with respect tothe filtration(Ft) is a continuous(Ft)-adapted real-valued process(W(t), t0) suchthat

W(0)=0;

for all0

s

t: W(t)

W(s) is independent ofFs;

for all0st: W(t) W(s) isN(0, t s)-distributed.1.2.2 Remark. Brownian motion can be constructed in different ways (Karatzas andShreve 1991), but the proof of the existence of such a process is in any case non-trivial.

We shall often consider a larger filtration(Ft) than the canonical filtration(FWt )

of Brownian motion in order to include random initial conditions. Given a Brownianmotion processW on a probability space(,F,P)with the canonical filtrationFt =(W(s), s t) and the random variable X0 on a different space (,F,P), wecan construct the product space with = , F = F F, P = P P suchthatW(t, , ) :=W(t, )andX0(

, ) :=X0 ()are independent andW is an

(Ft)-Brownian motion forFt = (X0; W(s), s t). Note that X0 isF0-measurablewhich always implies thatX0 andWare independent.

Our aim here is to construct the integralt0

Y(s) dW(s) with Brownian motion asintegrator and a fairly general class of stochastic integrands Y.

1.2.3 Definition. LetVbe the class of real-valued stochastic processes(Y(t), t0)that are adapted, measurable and that satisfy

YV :=

0

E

Y(t)2

dt1/2


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1.2. The Ito Integral 7

A processYV is called simple if it is of the form

Y(t, ) =

i=0

i()1[ti,ti+1)(t),

with an increasing sequence(ti)i0 andFti-measurable random variablesi.

For such simple processes YV we naturally define 0

Y(t) dW(t) :=i=0

i(W(ti+1) W(ti)). (1.2.1)

1.2.4 Proposition. The right hand side in (1.2.1) converges in L2(P), hence the

integral 0 Y(t) dW(t) is aP-almost surely well defined random variable. Moreoverthe following isometry is valid for simple processesY:E

0

Y(t) dW(t)2

=Y2V.

Proof. We show that the partial sums Sk :=ki=0 i(W(ti+1)W(ti)) form a Cauchy

sequence in L2(P). Let k l, then by the independence and zero mean property ofBrownian increments we obtain

ESl Sk2

=l

i=k+1Ei(W(ti+1) W(ti))2

+ 2

k+1i


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The main idea for extending the Ito integral to general integrands in Vis to showthat the simple processes lie dense in V with respect to theV-seminorm and touse the isometry to define the integral by approximation.1.2.5 Proposition. For any processY V there is a sequence of simple processes(Yn)n1 inV withlimnY YnV = 0.Proof. We proceed by relaxing the assumptions on Ystep by step:

1. Y is continuous,|Y(t)| K fortT and Y(t) = 0 fortT: Settni := in anddefine

Yn(t) :=Tn1i=0

Y(ti)1[tni,tni+1)(t).

Then Yn is clearly a simple process in V and by the continuity of Y(t) theprocesses Yn converge to Y pointwise for all (t, ). SinceYn2VT K2 holds,the dominated convergence theorem implies limnY YnV = 0.

2.|Y(t)| KfortT andY(t) = 0 fortT,T N:Ycan be approximated bycontinuous functionsYn inV with these properties (only Treplaced by T+ 1).For this suppose that h : [0, ) [0, ) is continuous, satisfies h(t) = 0 fort1 and h= 1. Forn Ndefine the convolution

Yn(t) :=

t0

Y(s) 1nh(n(t s)) ds.

Then Yn is continuous, has support in [0, T+ 1n ] and satisfies|Yn(t)| K for

all . Moreover, Yn(t) is Ft-adapted so thatYnVholds. Real analysis showsthat

(Yn Y)2 0 holds for n and all, so the assertion follows again

by dominated convergence.

3. YV arbitrary: The processes

Yn(t) :=

0, tnY(t), |Y(t)| n,t < nn, Y(t)> n

n, Y(t)0

p P

sup0nN

|Xn| E[|XN|p],

and for everyp > 1

E

sup0nN

|Xn|p p

p 1pE[|XN|p].

Proof. Introduce the stopping time := inf{n | |Xn| } N. Since (|Xn|p) is asubmartingale the optional stopping theorem gives

E[|XN|p] E[|X|p]p P

supn

|Xn|

+ E[|XN|p1{supn|Xn|


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which proves the first part. Moreover, we deduce from this inequality for any K >0andp >1

E

supn

|Xn| Kp] = E K0

pp11{supn|Xn|} d

K0

pp2 E[|XN|1{supn|Xn|}] d

=pE|XN|

supn|Xn|K0

p2 d

= p

p 1 E|XN|(sup

n|Xn| K)p1

.

By Holders inequality,

Esupn

|Xn| Kp] pp 1 E

supn

|Xn| Kp(p1)/p E[|XN|p]1/p,which after cancellation and taking the limit K yields the asserted momentbound.

1.2.10 Corollary. (DoobsLp-inequality) If(X(t),Ft)tI is a right-continuous mar-tingale indexed by a subintervalI R, then for anyp >1

E

suptI

|X(t)|p1/p pp 1suptI E[|X(t)|

p]1/p.

Proof. By the right-continuity ofXwe can restrict the supremum on the left to acountable subset D I. This countable set D can be exhausted by an increasingsequence of finite sets Dn D with

nDn = D. Then the supremum over Dn

increases monotonically to the supremum over D, the preceding theorem applies foreachDn and the monotone convergence theorem yields the asserted inequality.

Be aware that DoobsLp-inequality is different for p = 1 (Revuz and Yor 1999, p.55).

1.2.11 Corollary. For any X V there exists a version of t0X(s) dW(s) that iscontinuous int, i.e. a continuous process(J(t), t0) with

PJ(t) = t0

X(s) dW(s)= 1 for allt0.Proof. Let (Xn)n1 be an approximating sequence for X of simple processes in V.

Then by definition In(t) :=t0Xn(s) dW(s) is continuous in t for all . Moreover,

In(t) is an Ft-martingale so that Doobs inequality and the Ito isometry yield theCauchy property

E

supt0

|Im(t) In(t)|24 sup

t0E[|Im(t) In(t)|2] = 4Xm Xn2V0


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form, n . By the Chebyshev inequality and the Lemma of Borel-Cantelli thereexist a subsequence (Inl)l1 andL() such thatP-almost surely

lL() supt0

|Inl+1(t) Inl(t)| 2l.

Hence with probability one the sequence (Inl(t))l1 converges uniformly and the limitfunctionJ(t) is continuous. Since for all t0 the random variables (Inl(t))l1 con-verge in probability to the integral I(t) =

t0X(s) dW(s), the random variables I(t)

andJ(t) must coincide for P-almost all .

In the sequel we shall consider onlyt-continuous versions of the stochastic integral.

1.2.4 Extension of the Ito integral

We extend the stochastic integral from processes in V to the more general class ofprocessesV.

1.2.12 Definition. LetV be the class of real-valued stochastic processes(Y(t), t0) that are adapted, measurable and that satisfy

P

0

Y(t)2 dt


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By localisation via the stopping times (n) one can infer the properties of theextended integral from Theorem 1.2.7. The last assertion of the following theorem is

proved in (Revuz and Yor 1999, Prop. IV.2.13).1.2.15 Theorem. The stochastic integral over integrands inV has the same prop-erties as that over integrands inV regarding linearity (Theorem 1.2.7(c,d)), measur-ability (1.2.7(f)) and existence of a continuous version (1.2.7(h)). However, it is onlya local(Ft)-martingale with quadratic covariation as in (1.2.7(i)).

Moreover, ifYV is left-continuous and is a partition of[0, t], then the finitesum approximations converge in probability: t

0

Y(s) dW(s) = lim||0

ti

Y(ti)(W(ti+1) W(ti)).

1.2.5 The Fisk-Stratonovich integral

For integrands Y Van alternative reasonable definition of the stochastic integralis by interpolation T

0

Y(t) dW(t) := lim||0

ti

12(Y(ti+1)) + Y(ti))(W(ti+1) W(ti)),

where denotes a partition of [0, T] with 0 = t0 < t1


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1.2.6 Multidimensional Case

1.2.19 Definition.

1. AnRm-valued (Ft)-adapted stochastic process W(t) = (W1(t), . . . , W m(t))T isanm-dimensional Brownian motion if each componentWi, i = 1, . . . , m, is aone-dimensional(Ft)-Brownian motion and all components are independent.

2. If Y is anRdm-valued stochastic process such that each component Yij, 1i d, 1 j m, is an element of V then the multidimensional Ito inte-gral

Y dW form-dimensional Brownian motionW is anRd-valued random

variable with components

0

Y(t) dW(t)i

:=mj=1

0

Yij(t) dWj(t), 1id.

1.2.20 Proposition. The Ito isometry extends to the multidimensional case such thatforRdm-valued processesX, Ywith components inV andm-dimensional BrownianmotionW

E

0

X(t) dW(t),

0

Y(t) dW(t)=

0

d

i=1m

j=1 E[Xij(t)Yij(t)] dt.Proof. The term in the brackets on the left hand side is equal to

di=1

mj=1

mk=1

0

Xij(t) dWj(t)

0

Yik(t) dWk(t)

and the result follows from the one-dimensional Ito isometry once the following claimhas been proved: stochastic integrals with respect to independent Brownian motions

are uncorrelated (attention: they may well be dependent!).For this let us consider two independent Brownian motions W1 and W2 and two

simple processesY1, Y2 inVon the same filtered probability space with

Yk(t) =i=0

ik()1[ti,ti+1)(t), k {1, 2}.

The common partition of the time axis can always be achieved by taking a common


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refinement of the two partitions. Then by the Fti-measurability ofik we obtain

E 0

Y1(t) dW1(t) 0

Y2(t) dW2(t)=

0ij


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from Taylors formula

F(t, X(t)) = F(0, X(0)) +

nk=1

F(tk, X(tk)) F(tk1, X(tk1))

=F(0, X(0)) +n1k=0

Ft tk+

FxX(tk)

+ 122Fx2

(X(tk))2 + o(tk) + O((tk)(X(tk))) + o((X(tk))

2)

,

where all derivatives are evaluated at (X(tk), tk) and where we have set tk =tk+1tk,X(tk) =X(tk+1) X(tk). If we now let the width of the partition|| tend to zero,we obtain by the continuity ofXand the construction of the Riemann integral

n1k=0

Ft tk

t0

Ft (X(s), s) ds

and by the identity X(tk) =tk+1tk

g(s) ds+tk+1tk

h(s) dW(s) and the constructionof the Ito integral

n1k=0

Fx

X(tk) t0

Fx

(X(s), s)g(s) ds +

t0

Fx

(X(s), s)h(s) dW(s)

with convergence in L2(P). Note that for the precise derivation of the stochasticintegral we have to consider as approximating integrands the processes

Y(s) =h(s)n1k=0

Fx

(tk, X(tk))1[tk,tk+1)(s), s[0, t].

The third term converges to the quadratic variation process, using that an absolutelycontinuous function has zero quadratic variation:

n1

k=02Fx2

(X(tk))2

t0

2Fx2

(X(s), s)h2(s) ds.

The remainder terms converge to zero owing to the finite variation of 0

g(s) dsand the finite quadratic variation of

0 h(s) dW(s), which implies that the respective

higher order variations vanish.

1.2.22 Theorem. For anRdm-valued process h with components in V and anadaptedRd-valued process(g(t), t0) with T0g(t) dt 0 set

X(t) :=

t0

g(s) ds +

t0

h(s) dW(s), t0,


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whereW is anm-dimensional Brownian motion. ThenY(t) =F(X(t), t),t0, witha functionF C2,1(RdR+,Rp) satisfies

Y(t) =Y(0) + t0

Ft

(s, X(s)) + DF(X(s), s)g(s)

+ 12

di,j=1

2Fxixj

(X(s), s) ml=1

hi,l(s)hj,l(s)

ds

+

t0

DF(X(s), s)h(s) dW(s), t0.

HereDF = (xiFj)1id,1jp denotes the Jacobian ofF.

1.2.23 Remark. Itos formula is best remembered in differential form

dF =Ft dt + Fx dX(t) + 12

Fxx dXt (one-dimensional).A rule of thumb for deriving also the multi-dimensional formula is to simplify theTaylor expansion by proceeding formally and then substitutingdtidtj =dtidWj(t) = 0anddWi(t)dWj(t) =i,jdt.

If the stochastic integrals on the right hand side in the two preceding theoremsare interpreted in the Fisk-Stratonovich sense and if h is constant, then the termsinvolving second derivatives do not appear in the corresponding formulae:

dF =Ft dt + Fxg dt + hFx

dW(t) (one-dimensional).

Note, however, that for non-constanth we should writedF = Ft dt+Fxg dt+FxhdW(t), where the last term is a Stratonovich integral with respect to the continuousmartingale

0 h(s) dW(s) instead of Brownian motion.

1.2.24 Problem.

1. Consider again Problem 1.2.7 and evaluatet0W(s) dW(s)by regardingY(t) =

W(t)2.

2. Show that X(t) = exp(W(t) + (a 22

)t), W a one-dimensional Brownian

motion, satisfies the linear Ito stochastic differential equationdX(t) =aX(t) dt + X(t) dW(t).

What would be the solution of the same equation in the Fisk-Stratonovitch in-terpretation?

3. SupposeW is an m-dimensional Brownian motion, m 2, started inx= 0.Consider the processY(t) =W(t) (Euclidean norm) and find an expression

for the differentialdY(t), assuming thatWdoes not hit zero.


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The Ito formula allows a rather simple proof of Levys martingale characterisationof Brownian motion.

1.2.25 Theorem. Let(M(t),Ft, t0) be a continuousRm-valued local martingalewithM(0) = 0 and cross-variationsMk, Mlt = kltfor1k, ldP-almost surely.Then(M(t), t0) is anm-dimensional(Ft)-Brownian motion.Proof. We only sketch the proof for m = 1 and proper martingales M, details canbe found in (Karatzas and Shreve 1991, Thm. 3.16); in particular the integrationtheory for general semimartingales. In order to show that Mhas independent normallydistributed increments, it suffices to show

E[exp(iu(M(t) M(s))) |Fs] = exp(u2(t s)/2), uR, ts0.

By Itos formula for general continuous semimartingales applied to real and imaginarypart separately we obtain

exp(iuM(t)) = exp(iuM(s)) + iu

ts

exp(iuM(v)) dM(v) 12

u2 ts

exp(iuM(v)) dv.

Due to|exp(iuM(v))| = 1 the stochastic integral is a martingale and the functionF(t) = E[exp(iu(M(t) M(s))), |Fs] satisfies

F(t) = 1 12u2

t

s

F(v) dv P -a.s.

This integral equation has the unique solution F(t) = exp(u2(t s)/2).


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Chapter 2

Strong solutions of SDEs

2.1 The strong solution concept

The first definition of a solution of a stochastic differential equation reflects the in-terpretation that the solution process Xat time t is determined by the equation andthe exogenous input of the initial condition and the path of the Brownian motion upto time t. Mathematically, this is translated into a measurability condition on Xt orequivalently into the smallest reasonable choice of the filtration to which X shouldbe adapted, see condition (a) below.

2.1.1 Definition. A strong solutionXof the stochastic differential equation

dX(t) =b(X(t), t) dt + (X(t), t) dW(t), t0, (2.1.1)

with b : RdR+ Rd, : RdR+ Rdm measurable, on the given probabilityspace(,F,P) with respect to the fixedm-dimensional Brownian motionW and theindependent initial condition X0 over this probability space is a stochastic process(X(t), t0) satisfying:

(a) Xis adapted to the filtration(Gt), whereG0t :=(W(s), 0st) (X0)and

Gt is the completion ofs>t G

0s withP-null sets;

(b) Xis a continuous process;

(c) P(X(0) =X0) = 1;

(d) P(t0b(X(s), s) + (X(s), s)2 ds 0;

(e) With probability one we have

X(t) =X(0) +

t0

b(X(s), s) ds +

t0

(X(s), s) dW(s), t0.


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24/56

2.2. Uniqueness 21

Already for deterministic differential equations examples of nonuniqueness arewell known. For instance, the differential equation x(t) =|x(t)| with 0< 0, c0 andu, v : [0, T]R+ be measurable functions. Ifu is bounded andv is integrable, then

u(t)c + t0

u(s)v(s) ds t[0, T]

implies

u(t)c exp t

0

v(s) ds

, t[0, T].

Proof. Suppose c >0 and set

z(t) :=c +

t0

u(s)v(s) ds, t[0, T].

Then u(t)z(t),z(t) is weakly differentiable and for almost all t

z(t)z(t)

= u(t)v(t)z(t)

v(t)

holds so that log(z(t))log(z(0)) + t0

v(s) ds follows. This shows that

u(t)z(t)c exp t

0

v(s) ds

, t[0, T].

Forc = 0 apply the inequality for cn> 0 with limn cn= 0 and take the limit.

2.2.3 Theorem. Suppose thatb and are locally Lipschitz continuous in the space

variable, that is, for all n N there is a Kn > 0 such that for all t 0 and allx, y Rd withx, y n

b(x, t) b(y, t) + (x, t) (y, t) Knx y

holds. Then strong uniqueness holds for equation (2.1.1).

Proof. Let two solutions X and X of (2.1.1) with the same initial condition X0 begiven on some common probability space (,F,P). We define the stopping timesn := inf{t > 0 | X(t) n} and n in the same manner for X, n N. Then


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22 Chapter 2. Strong solutions of SDEs

n :=nnconverges P-almost surely to infinity. The differenceX(tn)X(tn)equalsP-almost surely

tn0

(b(X(s), s) b(X(s), s)) ds + tn0

((X(s), s) (X(s), s)) dW(s).

We conclude by the Ito isometry and Cauchy-Schwarz inequality:

E[X(t n) X(t n)2]

2E tn

0

b(X(s), s) b(X(s), s) ds2

+ 2E tn

0

(X(s), s) (X(s), s)2 ds

2T K2n t0

E[X(s n) X(s n)2] ds + 2K2n t0

E[X(s n) X(s n)2] ds.

By Gronwalls inequality we conclude P(X(tn) = X(tn)) = 1 for all n Nandt[0, T]. Lettingn, T , we see that X(t) =X(t) holdsP-almost surely forallt0 and by Remark 2.1.4 strong uniqueness follows.2.2.4 Remark. In the one-dimensional case strong uniqueness already holds forHolder-continuous diffusion coefficient of order1/2, see (Karatzas and Shreve 1991,Proposition 5.2.13) for more details and refinements.

2.3 Existence

In the deterministic theory differential equations are usually solved locally aroundthe initial condition. In the stochastic framework one is rather interested in globalsolutions and then uses appropriate stopping in order to solve an equation up to somerandom explosion time. To exclude explosions in finite time, the linear growth of thecoefficients suffices. The standard example for explosion is the ODE

x(t) =x(t)2, t0, x(0)= 0.Its solution is given byx(t) = 1/(x10 t) which explodes forx0> 0 andtx10 . Notealready here that with the opposite sign x(t) =x(t)2 the solutionx(t) =x(0)/(1+t)exists globally. Intuitively, the different behaviour is clear because in the first case xgrows the faster the further away from zero it is (positive feedback), while in the

second case x monotonically converges to zero (negative feedback).We shall first establish an existence theorem under rather strong growth and

Lipschitz conditions and then later improve on that.

2.3.1 Theorem. Suppose that the coefficients satisfy the global Lipschitz and lineargrowth conditions

b(x, t) b(y, t) + (x, t) (y, t) Kx y x, y Rd, t0 (2.3.1)b(x, t) + (x, t) K(1 + x) xRd, t0 (2.3.2)


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2.3. Existence 23

with some constantK >0. Moreover, suppose that on some probability space(,F,P)there exists anm-dimensional Brownian motionW and an initial conditionX0 with

E[X02

] 0 the moment bound

E[X(t)2]C(1 + E[X02])eCt2, t0.Proof. As in the deterministic case we perform successive approximations and applya Banach fixed point argument (Picard-Lindelof iteration). Define recursively

X0(t) :=X0, t0 (2.3.3)

Xn+1(t) :=X0+ t

0

b(Xn(s), s) ds + t

0

(Xn(s), s) dW(s), t

0. (2.3.4)

Obviously, the processesXn are continuous and adapted to the filtration generated byX0 andW. Let us fix some T >0. We are going to show that for arbitrary t[0, T]

E

sup0st

Xn+1(s) Xn(s)2C1 (C2t)

n

n! (2.3.5)

holds with suitable constantsC1, C2> 0 independent oft and n and C2 = O(T). Letus see how we can derive the theorem from this result. From Chebyshevs inequalitywe obtain

P sup0sTXn+1(s) Xn(s)> 2n14C1 (4C2T)nn!The term on the right hand side is summable over n, whence by the Borel-CantelliLemma we conclude

P

for infinitely manyn: sup0sT

Xn+1(s) Xn(s)> 2n1

= 0.

Therefore, by summation supm1sup0sTXn+m(s)Xn(s) 2n holds for alln N() with some P-almost surely finite random index N(). In particular, therandom variables Xn(s) form a Cauchy sequence P-almost surely and converge tosome limit X(s), s

[0, T]. Obviously, this limiting process Xdoes not depend on

Tand is thus defined on R+. Since the convergence is uniform over s [0, T], thelimiting process X is continuous. Of course, it is also adapted by the adaptednessofXn. Taking the limit n in equation (2.3.4), we see that X solves the SDE(2.1.1) up to time Tbecause of

sup0sT

b(Xn(s), s) b(XT(s), s) K sup0sT

Xn(s) XT(s) 0 (in L2(P))

E(Xn(), ) (X(), )2V([0,T])]K2T sup

0sTE[Xn(s) X(s)2]0.


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Since T > 0 was arbitrary, the equation (2.1.1) holds for all t 0. From estimate(2.3.5) and the asymptotic bound C2= O(T) we finally obtain by summation over n

and putting T =t the asserted estimate onE[X(t)2

].It thus remains to establish the claimed estimate (2.3.5), which follows essentially

from Doobs martingale inequality and the type of estimates used for proving Theorem2.2.3. Proceeding inductively, we infer from the linear growth condition that (2.3.5)is true for n= 0 with some C1 >0. Assuming it to hold for n1, we obtain with aconstant D >0 from Doobs inequality:

E

sup0st

Xn+1(s) Xn(s)2

2E

sup0st

s

0

b(Xn(u), u) b(Xn1(u), u) du2

+ 2E

sup0st

s0

(Xn(u), u) (Xn1(u), u) dW(u)22K2t

t0

E[Xn(u) Xn1(u)2] du + 2DK2 t0

E[Xn(u) Xn1(u)2] du

(2K2T C1+ 2DK2)Cn12tn

n!.

The choice C2 = 2K2(T C1+ D)/C1 = O(T) thus gives the result.

The last theorem is the key existence theorem that allows generalisations into

many directions. The most powerful one is essentially based on conditions such thata solution Xexists locally andX(t)2 remains bounded for all t0 (L(x) =x2 isa Lyapunov function). Our presentation follows Durrett (1996).

2.3.2 Lemma. SupposeX1 andX2 are adapted continuous processes withX1(0) =X2(0) andE[X1(0)2]


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2.3. Existence 25

2.3.3 Theorem. Suppose the drift and diffusion coefficientsb and are locally Lip-schitz continuous in the space variable and satisfy for someB0

2x, b(x, t) + trace((x, t)(x, t)T)B(1 + x2), xRd, t0,then the stochastic differential equation (2.1.1) has a strong solution for any initialconditionX0 satisfyingE[X02]0define coefficient functions bR, R such that

bR(x) =

b(x), x R,0, x 2R, and R(x) =

(x), x R,0, x 2R,

andbRandRare interpolated forx (R, 2R) in such a way that they are Lipschitzcontinuous in the state variable. Then let XR be the by Theorem 2.3.1 unique strongsolution to the stochastic differential equation with coefficients bR andR. Introducethe stopping time R := inf{t0 | XR(t) R}. Then by Lemma 2.3.2 XR(t) andXS(t) coincide for tmin(R, S) and we can define

X(t) :=XR(t) fortR.The processXwill be a strong solution of the stochastic differential equation (2.1.1)if we can show limR R =P-almost surely.

Put(x) = 1 + x2. Then Itos formula yields for any t,R >0eBt(XR(t)) (XR(0))

=B t0

eBs(XR(s)) ds +di=1

t0

eBs2XR,i(s) dXR,i(s)

+1

2

di=1

t0

eBs2dj=1

ij(XR(s), s)2 ds

= local martingale

+

t0

eBsB(XR(s)) + 2x, bR(XR(s), s) + trace(R(XR(s), s)TR(XR(s), s))

ds.

Our assumption implies that (eB(tR)(XR(t R)))t0 is a supermartingale bythe optional stopping theorem. We conclude

E[(X0)] E

eB(tR)(XR(t R))

= E

eB(tR)(X(t R))

eBt P(Rt) minx=R

(x).

Because of limx (x) = we have limR P(R t) = 0. Since the events({Rt})R>0 decrease, there exists for all t >0 andP-almost all an indexR0 suchthatR()t for all RR0, which is equivalent to R P-almost surely.


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2.4 Explicit solutions

2.4.1 Linear EquationsIn this paragraph we want to study the linear or affine equations

dX(t) =

A(t)X(t) + a(t)

dt + (t) dW(t), t0. (2.4.1)Here,A is a dd-matrix,ais a d-dimensional vector and is a dm-dimensionalmatrix, where all objects are determinisic as well as measurable and locally boundedin the time variable. As usual, W is an m-dimensional Brownian motion and X ad-dimensional process.

The corresponding deterministic linear equation

x(t) =A(t)x(t) + a(t), t0, (2.4.2)has for every initial condition x0 an absolutely continuous solution x, which is givenby

x(t) = (t)

x0+

t0

1(s)a(s) ds

, t0,

where is the so-called fundamental solution. This means that solves the matrixequation

(t) =A(t)(t), t0, with (0) = Id .In the case of a matrix Athat is constant in time, the fundamental solution is given

by

(t) =eAt :=k=0

(tA)k

k! .

2.4.1 Proposition. The strong solutionXof equation (2.4.1)with initial conditionX0 is given by

X(t) = (t)

X0+

t0

1(s)a(s) ds +

t0

1(s)(s) dW(s)

, t0.

Proof. Apply Itos formula.

2.4.2 Problem.

1. Show that the function (t) := E[X(t)] under the hypothesisE[|X(0)|]


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2.4. Explicit solutions 27

2.4.2 Transformation methods

We follow the presentation by Kloeden and Platen (1992) and consider scalar equa-

tions that can be solved explicitly by suitable transformations.Consider the scalar stochastic differential equation

dX(t) = 12b(X(t))b(X(t)) dt + b(X(t)) dW(t), (2.4.3)

where b :R R is continously differentiable and does not vanish and W is a one-dimensional Brownian motion. This equation is equivalent to the Fisk-Stratonovichequation

dX(t) =b(X(t)) dW(t).Define

h(x) := xc 1b(y)dy for some c R .Then X(t) := h1(W(t) + h(X0)), where h

1 denotes the inverse ofh which existsby monotonicity, solves the equation (2.4.3). This follows easily from (h1)(W(t) +h(X0)) =b(X(t)) and (h

1)(W(t) + h(X0)) =b(X(t))b(X(t)).

2.4.3 Example.

1. (geometric Brownian motion)dX(t) = 2

2X(t) dt+X(t) dW(t)has the solutionX(t) =X0exp(W(t)).

2. The choiceb(x) =

|x

| for,

R corresponds formally to the equation

dX(t) = 12

2|X(t)|21 sgn(X(t)) dt + |X(t)| dW(t).For 2 also the trivial processX(t) = 0 isa solution.


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3. The equation

dX(t) =

a2 sin(X(t))cos3(X(t)) dt + a cos2(X(t)) dW(t)

has for X0 (2 , 2 ) the solution X(t) = arctan(aW(t) + tan(X0)), whichremains contained in the interval (2 , 2 ). This can be explained by the factthat forx=

2the coefficients vanish and for valuesx close to this boundary

the drift pushes the process towards zero more strongly than the diffusion partcan possibly disturb.

4. The equation

dX(t) =a2X(t)(1 + X(t)2) dt + a(1 + X(t)2) dW(t)

is solved byX(t) = tan(aW(t) + arctan X0) and thus explodesP-almost surelyin finite time.

The transformation idea allows certain generalisations. With the same assump-tions on b and the same definition ofh we can solve the equation

dX(t) =

b(X(t)) +1

2b(X(t))b(X(t))

dt + b(X(t)) dW(t)

byX(t) =h1(t + W(t) + h(X0)). Equations of the type

dX(t) =

h(X(t))b(X(t)) +

1

2b(X(t))b(X(t))

dt + b(X(t)) dW(t)

are solved byX(t) =h1(eth(X0) + et t

0es dW(s)).

Finally, we consider forn N, n2, the equationdX(t) = (aX(t)n + bX(t)) dt + cX(t) dW(t).

WritingY(t) =X(t)1n we obtain

dY(t) = (1 n)X(t)n dX(t) + 12

(1 n)(n)X(t)n1c2X2(t) dt= (1 n)(a + (b c2

2n)Y(t))dt + (1 n)cY(t) dW(t).

Hence, Yis a geometric Brownian motion and we obtain after transformation for allX0= 0

X(t) =e(bc2

2 )t+cW(t)

X1n0 + a(1 n)

t0

e(n1)(bc2

2)s+c(n1)W(s) ds

1/(1n).

In addition to the trivial solution X(t) = 0 we therefore always have a nonnegativeglobal solution in the case X00 and a0. For odd integers n and a0 a globalsolution exists for any initial condition, cf. Theorem 2.3.3. In the other cases it iseasily seen that the solution explodes in finite time.


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Chapter 3

Weak solutions of SDEs

3.1 The weak solution concept

We start with the famous example of H. Tanaka. Consider the scalar SDE

dX(t) = sgn(X(t)) dW(t), t0, X(0) = 0, (3.1.1)

where sgn(x) = 1(0,)(x)1(,0](x). Any adapted process X satisfying (3.1.1) isa continuous martingale with quadratic variationXt = t. Levys Theorem 1.2.25implies that Xhas the law of Brownian motion. If X satisfies this equation, thenso doesX, since the Lebesgue measure of{t [0, T] | X(t) = 0} vanishes almostsurely for any Brownian motion. Hence strong uniqueness cannot hold.

We now invert the roles of X and W, for equation (3.1.1) obviously impliesdW(t) = sgn(X(t)) dX(t). Hence, we take a probability space (,F,P) equippedwith a Brownian motion Xand consider the filtration (FXt )t0 generated by X andcompleted underP. Then we define the process

W(t) :=

t0

sgn(X(s)) dX(s), t0.

Wis a continuous (FXt )-adapted martingale with quadratic variationWt = t, hencealso an (FXt )-Brownian motion. The couple (X, W) then solves the Tanaka equation.

However, X is not a strong solution because the filtration (FWt )t0 generated by Wand completed underP satisfies FWt F

Xt as we shall see.

For the proof let us take a sequence (fn) of continuously differentiable functions onthe real line that satisfyfn(x) = sgn(x) for|x| 1n and|fn(x)| 1, fn(x) =fn(x)for allxR. If we setFn(x) =

x0 fn(y) dy, thenFnC2(R) and limn Fn(x) =|x|

holds uniformly on compact intervals. By Itos formula for any solution Xof (3.1.1)

Fn(X(t)) t0

fn(X(s))dX(s) =1

2

t0

fn(X(s))ds, t0,


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30 Chapter 3. Weak solutions of SDEs

follows and by Lebesgues Theorem the left hand side converges in probability forn to|X(t)|

t

0sgn(X(s))dX(s) =|X(t)|W(t). By symmetry,fn(x) =fn(|x|)

and we have for t0 P-almost surelyW(t) =|X(t)| lim

n

1

2

t0

fn(|X(s)|)ds.

Hence, FWt F|X|t holds with obvious notation. The event{X(t)>0}has probability

12

> 0 and is not F|X|t -measurable. Therefore F

Xt \ F|X|t is non-void and FWt FXt

holds for any solution X, which is thus not a strong solution in our definition. Notethat the above derivation would be clearer with the aid of Tanakas formula and theconcept of local time.

3.1.1 Definition. A weak solution of the stochastic differential equation (2.1.1) is atriple(X, W), (,F,P), (Ft)t0 where

(a) (,F,P) is a probability space equipped with the filtration (Ft)t0 that satisfiesthe usual conditions;

(b) X is a continuous, (Ft)-adaptedRd-valued process andW is anm-dimensional(Ft)-Brownian motion on the probability space;

(c) conditions (d) and (e) of Definition 2.1.1 are fulfilled.

The distributionPX(0) ofX(0) is called initial distribution of the solutionX.

3.1.2 Remark. Any strong solution is also a weak solution with the additional filtra-tion propertyFXt F

Wt (X(0)). The Tanaka equation provides a typical example

of a weakly solvable SDE that has no strong solution.

3.1.3 Definition. We say that pathwise uniqueness for equation (2.1.1)holds when-ever two weak solutions(X, W), (,F,P), (Ft)t0 and(X, W), (,F,P), (Ft)t0 ona common probability space with a common Brownian motion with respect to both

filtrations (Ft) and (Ft), and withP(X(0) = X

(0)) = 1 satisfyP( t 0 : X(t) =X(t)) = 1.

3.1.4 Definition. We say that uniqueness in law holds for equation(2.1.1)whenevertwo weak solutions (X, W), (,F,P), (Ft)t0 and (X, W), (,F,P), (Ft)t0 withthe same initial distribution have the same law, that isP(X(t1) B1, . . . , X (tn)Bn) =P(X(t1)B1, . . . , X (tn)Bn) holds for allnN, t1, . . . , tn > 0 and BorelsetsB1, . . . , Bn.

3.1.5 Example. For the Tanaka equation pathwise uniqueness fails becauseX andX are at the same time solutions. We have, however, seen thatX must have thelaw of a Brownian motion and thus uniqueness in law holds.


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3.2. The two concepts of uniqueness 31

3.2 The two concepts of uniqueness

Let us discuss the notion of pathwise uniqueness and of uniqueness in law in somedetail. When we consider weak solutions we are mostly interested in the law of thesolution process so that uniqueness in law is usually all we require. However, as weshall see, the concept of pathwise uniqueness is stronger than that of uniqueness inlaw and if we reconsider the proof of Theorem 2.2.3 we immediately see that we havenot used the special filtration properties of strong uniqueness and we obtain:

3.2.1 Theorem. Suppose thatb and are locally Lipschitz continuous in the spacevariable, that is, for all n N there is a Kn > 0 such that for all t 0 and allx, y Rd withx, y n

b(x, t)

b(y, t)

+

(x, t)

(y, t)

Kn

x

y

holds. Then pathwise uniqueness holds for equation (2.1.1).

The same remark applies to Example 2.2.1. As Tanakas example has shown,pathwise uniqueness can fail when uniqueness in law holds. It is not clear, though,that the converse implication is true.

3.2.2 Theorem. Pathwise uniqueness implies uniqueness in law.

Proof. We have to show that two weak solutions (Xi, Wi), (i,Fi,Pi), (Fit), i = 1, 2on possibly different filtered probability spaces agree in distribution. The main ideais to define two weak solutions with the same law on a common space with the same

Brownian motion and to apply the pathwise uniqueness assumption. To this end weset

S:= RdC(R+,Rm) C(R+,Rd), S = Borel -field ofSand consider the image measures

Qi(A) := Pi((Xi(0), Wi, Xi)A), A S, i= 1, 2.Since Xi(t) is by definition F

it-measurable, Xi(0) is independent ofWi under Pi. If

we call the law of Xi(0) under Pi (which by assumption does not depend on i),we thus have that the product measure Wis the law of the first two coordinates(Xi(0), Wi) under Pi, where W denotes the Wiener measure. Since C(R+,Rk) is aPolish space, a regular conditional distribution (Markov kernel) Ki of Xi under Pigiven (Xi(0), Wi) exists (Karatzas and Shreve 1991, Section 5.3D) and we may writefor Borel sets F RdC(R+,Rm), GC(R+,Rd)

Qi(F G) =F

Ki(x0, w; G) (dx0)W(dw).

Let us now define

T =S C(R+,Rd), T= Borel-field ofT


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and equip this space with the probability measure

Q(d(x

0, w , y

1, y

2)) =K

1(x

0, w; dy

1)K

2(x

0, w; dy

2)(dx

0)W

(dw).

Finally, denote by T the completion ofTunderQ and consider the filtrations

Tt = ((x0, w(s), y1(s), y2(s)), st)

and its Q-completion Tt and its right-continuous version T =

s>t T

s . Then the

projection on the first coordinate has under Q the law of the initial distribution ofXi and the projection on the second coordinate is underQ an Tt -Brownian motion(recall Remark 2.1.2). Moreover, the distribution of the projection (w, yi) under Qis the same as that of (Wi, Xi) under Pi such that we have constructed two weaksolutions on the same probability space with the same initial condition and the sameBrownian motion.

Pathwise uniqueness now implies Q({(x0, w , y1, y2) T|y1 = y2}) = 1. Thisentails

P1((W1, X1)A) = Q((w, y1)A) = Q((w, y2)A) = P2((W2, X2)A).

The same methodology allows to prove the following, at a first glance ratherstriking result.

3.2.3 Theorem. The existence of a weak solution and pathwise uniqueness imply theexistence of a strong solution on any sufficiently rich probability space.

Proof. See (Karatzas and Shreve 1991, Cor. 5.3.23).

3.3 Existence via Girsanovs theorem

The Girsanov theorem is one of the main tools of stochastic analysis. In the theory ofstochastic differential equations it often allows to extend results for a particular equa-tion to those with more general drift coefficients. Abstractly seen, a Radon-Nikodymdensity for a new measure is obtained, under which the original process behavesdifferently. We only work in dimension one and start with a lemma on conditionalRadon-Nikodym densities.

3.3.1 Lemma. Let(,F,P)be a probability space, H Fbe a sub--algebra andfL1(P)be a density, that is nonnegative and integrating to one. Then a new probabilitymeasureQ onFis defined byQ(d) =f()P(d)and for anyF-measurable randomvariableX withEQ[|X|]


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3.3. Existence via Girsanovs theorem 33

3.3.2 Remark. In the unconditional case we obviously have

EQ[X] = X dQ = Xf dP = EP[Xf].Proof. We show that the left-hand side is a version of the conditional expectation onthe right. Since it is obviously H-measurable, it suffices to verify

H

EQ[X|H]EP[f|H] dP =H

Xf dP =H

X dQ H H.

By the projection property of conditional expectations we obtain

EP[1HEQ[X|H]EP[f

|H]] = EP[1HEQ[X

|H]f] = EQ[1HEQ[X

|H]] = EQ[1HX],

which is the above identity.

3.3.3 Lemma. Let ((t), 0 t T) be an (Ft)-adapted process with1tT V.Then

M(t) := exp t0

(s) dW(s) 12

t0

2(s) ds

, 0tT ,

is an(Ft)-supermartingale. It is a martingale if and only ifE[M(T)] = 1 holds.

Proof. If we apply Itos formula to M, we obtain

dM(t) =(t)M(t) dW(t), 0tT .

Hence, M is always a nonnegative local P-martingale. By Fatous lemma for condi-tional expectations we infer that Mis a supermartingale and a proper martingale ifand only ifEP[M(T)] = EP[M(0)] = 1.

3.3.4 Lemma. M is a martingale ifsatisfies one of the following conditions:

1. is uniformly bounded;

2. Novikovs condition:

E

exp1

2

T0

2(t) dt


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Proof. By the previous proof we know that M solves the linear SDE dM(t) =(t)M(t) dW(t) with M(0) = 1. Since (t) is uniformly bounded, the diffu-sion coefficient satisfies the linear growth and Lipschitz conditions and we couldmodify Theorem 2.3.1 to cover also stochastic coefficients and obtain equally thatsup0tTE[M(t)

2] is finite. This implies M1[0,T]V and Mis a martingale.Alternatively, we proveM1[0,T]Vby hand: Ifis uniformly bounded by some

K >0, then we have for any p >0 and any partition 0 =t0t1 tn= t

E

expp

ni=1

(ti1)(W(ti) W(ti1))

= E

exp

pn1

i=1(ti1)(W(ti) W(ti1))

E[exp(p(tn1)(W(tn) W(tn1)) |Ftn1]

= E

exp

pn1i=1

(ti1)(W(ti) W(ti1))

exp(p2(tn1)2(tn tn1)

E

exp

pn1i=1

(ti1)(W(ti) W(ti1))

exp(p2K2(tn tn1)

exp ni=1

p2K2(ti ti1)

= exp(p2K2t).

This shows that the random variables expni=1 (ti1)(W(ti)W(ti1)) are uni-formly bounded in any Lp(P)-space and thus uniformly integrable. Since by tak-ing finer partitions these random variables converge to exp(

t0

(s) dW(s)) in P-probability, we infer thatM(t) has finite expectation and even moments of all orders.

Consequently,T0 E[((t)M(t))2] dt is finite and Mis a martingale.

For the sufficency of Novikovs and Kazamakis condition we refer to (Liptser andShiryaev 2001) and the references and examples (!) there.

3.3.5 Theorem. Let(X(t), 0tT)be a stochastic (Ito) process on(,F, (Ft),P)satisfying

X(t) = t0

(s) ds + W(t), 0tT ,

with a Brownian motion W and a process 1tT V. If is such that M is amartingale, then (X(t), 0 t T) is a Brownian motion under the measureQ on(,F, (Ft)) defined byQ(d) =M(T, )P(d).

Proof. We use Levys characterisation of Brownian motion from Theorem 1.2.25.SinceMis a martingale, M(T) is a density andQ is well-defined.


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3.3. Existence via Girsanovs theorem 35

We put Z(t) =M(t)X(t) and obtain by Itos formula (or partial integration)

dZ(t) =M(t) dX(t) + X(t) dM(t) + dM, X

t

=M(t)

(t) dt + dW(t) X(t)(t) dW(t) (t)dt=M(t)(1 X(t)(t)) dW(t).

This shows thatZis a local martingale. IfZis a martingale, then we accomplish theproof using the preceding lemma:

EQ[X(t) |Fs] = EP[M(t)X(t) |Fs]EP[M(t) |Fs] = Z(s)

M(s)=X(s), st,

implies thatX is aQ-martingale which by its very definition has quadratic variation

t. Hence, Xis a Brownian motion underQ.IfZis only a local martingale with associated stopping times (n), then the aboverelation holds for the stopped processesXn(t) =X(t n), which shows that X is alocalQ-martingale and Levys theorem applies.

3.3.6 Proposition. Suppose X is a stochastic process on (,F, (Ft),P) satisfyingfor someT >0 and measurable functionsb and

dX(t) =b(X(t), t) dt + (X(t), t) dW(t), 0tT , X(0) =X0.Assume further thatu(x, t) :=c(x, t)/(x, t), c measurable, is such that

M(t) = exp t0

u(X(s), s) dW(s) 1

2 t0 u2(X(s), s) ds, 0tT ,is an(Ft)-martingale.

Then the stochastic differential equation

dY(t) = (b(Y(t), t) + c(Y(t), t)) dt + (Y(t), t) dW(t), 0tT , Y(0) =X0,(3.3.1)

has a weak solution given by((X,W), (,F,Q), (Ft)) for theQ-Brownian motion

W(t) :=W(t) +

t0

u(X(s), s) ds, t0,

and the probabilityQ given byQ(d) :=M(T, )P(d).

3.3.7 Remark. Usually, the martingale(M(t), t0) is not closable whence we areled to consider stochastic differential equations for finite time intervals.

The martingale condition is for instance satisfied if is bounded away from zeroandc is uniformly bounded. Putting(x, t) = 1 andb(x, t) = 0we have weak existence

for the equationdX(t) =c(X(t), t) dt+dW(t) ifc is Borel-measurable and satisfiesa linear growth condition in the space variable, but without continuity assumption(Karatzas and Shreve 1991, Prop. 5.36).


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Proof. From Theorem 3.3.5 we infer that

W is aQ-Brownian motion. Hence, we can

write

dX(t) = b(X(t), t) (X(t), t)u(X(t), t) dt + (X(t), t) dW(t),which by definition ofu shows that (X,W) solves underQ equation (3.3.1).

The Girsanov Theorem also allows statements concerning uniqueness in law. Thefollowing is a typical version, which is proved in (Karatzas and Shreve 1991, Prop.5.3.10, Cor 5.3.11).

3.3.8 Proposition. Let two weak solutions((Xi, Wi), (i,Fi,Pi), (Fit)), i= 1, 2, of

dX(t) =b(X(t), t) dt + dW(t), 0

t

T ,

with b : RR+ R measurable be given with the same initial distribution. IfPi(T0|b(Xi(t), t)|2 dt


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Chapter 4

The Markov properties

4.1 General facts about Markov processes

Let us fix the measurable space (state space) (S,S) and the filtered probability space(,F,P; (Ft)t0) until further notice. We present certain notions and results concern-ing Markov processes without proof and refer e.g. to Kallenberg (2002) for furtherinformation. We specialise immediately to processes in continuous time and later onalso to processes with continuous trajectories.

4.1.1 Definition. An S-valued stochastic process (X(t), t 0) is called Markovprocess ifX is(Ft)-adapted and satisfies

0st, B S : P(X(t)B |Fs) = P(X(t)B | X(s)) P -a.s.In the sequel we shall always suppose that regular conditional transition probabil-

ities (Markov kernels)s,t exist, that is for all st the functionss,t : SS R aremeasurable in the first component and probability measures in the second componentand satisfy

s,t(X(s), B) = P(X(t)B | X(s)) = P(X(t)B |Fs) P -a.s. (4.1.1)4.1.2 Lemma. The Markov kernels(s,t)satisfy the Chapman-Kolmogorov equation

s,u(x, B) = St,u(y, B) s,t(x,dy) 0stu, xS, B S.4.1.3 Definition. Any family of regular conditional probabilities (s,t)st satisfyingthe Chapman-Kolmogorov equation is called a semigroup of Markov kernels. The ker-nels (or the associated process) are called time homogeneous ifs,t = 0,ts holds. Inthis case we just writets.

4.1.4 Theorem. For any initial distribution auf (S,S) and any semigroup ofMarkov kernels(s,t)there exists a Markov processX such thatX(0)is-distributedand equation (4.1.1) is satisfied.


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38 Chapter 4. The Markov properties

IfSis a metric space with Borel-algebraS and if the process has a continuousversion, then the process can be constructed on the path space =C(R+, S) with its

Borel -algebraB and canonical right-continuous filtrationFt =s>t (X(u), us), where X(u, ) := (u) are the coordinate projections. The probability measureobtained is calledPand it holds

P=S

Px(A) (dx), A B,

withPx:= Px.

For the formal statement of the strong Markov property we introduce the shiftoperator t that induces a left-shift on the function space .

4.1.5 Definition. The shift operator t on the canonical space is given by t :, t() =(t +) for allt0.4.1.6 Lemma.

1. t is measurable for allt0.2. For (Ft)-stopping times and the random time := + is again an

(Ft)-stopping time.

4.1.7 Theorem. LetXbe a time homogeneous Markov process and letbe an(Ft)-stopping time with at most countably many values. Then we have for allxS

Px(X A |F) = PX()(A) Px-a.s. AB. (4.1.2)IfXis the canonical process on the path space, then this is just an identity concerningthe image measure under()():Px( |F) ()1 = PX().4.1.8 Definition. A process X satisfying (4.1.2) for any finite (or equivalentlybounded) stopping time is called strong Markov.

4.1.9 Remark. The strong Markov property entails the Markov property by setting=t andA={X(s)B} for someB S in (4.1.2).

4.2 The martingale problemWe specify now to the state space S = Rd. As before we work on the path space =C(R+,Rd) with its Borel -algebra B.

4.2.1 Definition. A probability measureP on the path space(,B) is a solution ofthe local martingale problem for(b, ) if

Mf(t) :=f(X(t)) f(X(0)) t0

Asf(X(s)) ds, t0,


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4.2. The martingale problem 39

where

Asf(x) :=1

2

d

i,j=1(T(x, s))ij2f

xixj(x) +

b(x, s), grad(f)(x)

,

b :RdR+ Rd, :RdR+ Rdm measurable, is a local martingale underPfor all functionsf CK(Rd,R).4.2.2 Remark. Ifb and are bounded, thenP even solves the martingale problem,

for whichMf is required to be a proper martingale.

4.2.3 Theorem. The stochastic differential equation

dX(t) =b(X(t), t) dt + (X(t), t) dW(t), t0

has a weak solution ((X, W), (,A

,P), (Ft)) if and only if a solution to the localmartingale problem (b, ) exists. In this case the law PX of X on the path spaceequals the solution of the local martingale problem.

Proof. For simplicity we only give the proof for the one-dimensional case, the multi-dimensional method of proof follows the same ideas.

1. Given a weak solution, Itos rule yields for any fCK(R)df(X(t)) = f(X(t)) dX(t) + 1

2f(X(t)) dXt

=f(X(t))(X(t), t) dW(t) + Atf(X(t)) dt.

Hence, Mf

is a local martingale; just note that (X()) V

is required forthe weak solution andf is bounded such that the stochastic integral is indeedwell defined and a local martingale under P. Of course, this remains true, whenconsidered on the path space under the image measurePX.

2. Conversely, let P be a solution of the local martingale problem and considerfunctionsfnCK(R) with fn(x) =x for|x| n. Then the standard stoppingargument applied to Mfn forn shows that

M(t) :=X(t) X(0) t0

b(X(s), s) ds, t0,

is a local martingale. Similarly approximatingg(x) =x2, we obtain that

N(t) :=X2(t) X2(0) t0

2(X(s), s) + b(X(s), s)2X(s) ds, t0,

is a local martingale. By Itos formula,dX2(t) = 2X(t)dX(t) + dXt holds andshows

N(t) =

t0

2X(s) dM(s) + Mt t0

2(X(s), s) ds, t0.


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ThereforeMtt0

2(X(s), s) dsis a continuous local martingale of boundedvariation. By (Revuz and Yor 1999, Prop. IV.1.2) it must therefore vanish

identically and dMt = 2

(X(t), t) dt follows. By the representation theoremfor continuous local martingales (Kallenberg 2002, Thm. 18.12) there exists aBrownian motion W such thatM(t) =

t0(X(s), s) dW(s) holds for all t0.

Consequently (X, W) solves the stochastic differential equation.

4.2.4 Corollary. A stochastic differential equation has a (in distribution) uniqueweak solution if and only if the corresponding local martingale problem is uniquelysolvable, given some initial distribution.

4.3 The strong Markov property

We immediately start with the main result that solutions of stochastic differentialequations are under mild conditions strong Markov processes. This entails that thesolutions are diffusion processes in the sense of Feller (Feller 1971).

4.3.1 Theorem. Letb: Rd Rd and: Rd Rdm be time-homogeneous measur-able coefficients such that the local martingale problem for(b, )has a unique solutionPx for all initial distributions x, x Rd. Then the family (Px) satisfies the strongMarkov property.

Proof. In order to state the strong Markov property we need that (Px)xRdare Markovkernels. Theorem 21.10 of Kallenberg (2002) shows by abstract arguments that xPx(B) is measurable for all B B.

We thus have to show

Px(X B |F) = PX()(B) Px-a.s.B B, bounded stopping time .

By the unique solvability of the martingale problem it suffices to show that the random(!) probability measureQ := Px(()1 |F) solvesPx-almost surely the martingaleproblem for (b, ) with initial distribution X(). Concerning the initial distribution

we find for any Borel set ARd

by the stopping time property of

Px(()1{ | (0)A} |F)() = Px({ | (())A} |F)()=1A((())

=1A(X((), ))

= PX((),)({ | (0)A}).

It remains to prove the local martingale property of Mf under Q, that is themartingale property ofMf,n(t) := Mf(tn) with n := inf{t 0 | Mf(t) n}.


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4.3. The strong Markov property 41

By its very definition Mf(t) is always Ft-measurable, so we prove that Px-almostsurely

F

Mf,n(t, )Q(d) =

F

Mf,n(s, )Q(d) F Fs, st.

By the separability of and the continuity ofMf,n it suffices to prove this identityfor countably manyF,sand t (Kallenberg 2002, Thm. 21.11). Consequently, we neednot worry aboutPx-null sets. We obtain

F

Mf,n(t, )Q(d) =

1F((

))Mf,n(t, ())Px(d |F)

= Ex[1()1FMf,n(t, ) |F].

Because ofMf,n(t, ) =Mf((t + ) n) withn:= n + , which is by Lemma

4.1.6 a stopping time, the process Mf,n(t, ) is a martingale under Px adapted to(Ft+)t0. Since ()

1Fis an element ofFs+, we conclude by optional stopping thatPx-almost surely

F

Mf,n(t, )Q(d) = Ex[1()1FEx[M

f,n(t, ) |Fs+] |F]= Ex[1()1FM

f,n(s + ) |F]=

FMf,n(s, )Q(d

).

Consequently, we have shown that with Px-probability oneQ solves the martingaleproblem with initial distribution X() and therefore equalsPX().

4.3.2 Example. A famous application is the reflection principle for Brownian motionW. By the strong Markov property, for any finite stopping time the process(W(t +)W(), t 0) is again a Brownian motion independent of F such that withb:= inf{t0 | W(t)b} for someb >0:

P0(bt) = P0(bt, W(t)b) + P0(bt, W(t)< b)= P0(W(t)

b) + P0(b

t, W(b+ (t

b))

W(b) b) and the stopping timeb has a distributionwith density

fb(t) = b

2t3eb

2/(2t), t0.

Because of{bt}={max0stW(t)b}we have at the same time determined thedistribution of the maximum of Brownian motion on any finite interval.


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4.4 The infinitesimal generator

We first gather some facts concerning Markov transition operators and their semi-group property, see Kallenberg (2002) or Revuz and Yor (1999).

4.4.1 Lemma. Given a family(t)t0 of time-homogeneous Markov kernels, the op-erators

Ttf(x) :=

f(y)t(x,dy), f :S R bounded, measurable,

form a semigroup, that isTt Ts= Tt+s holds for allt, s0.Proof. Use the Chapman-Kolmogorov equation.

We now specialise to the state space S= Rd with its Borel-algebra.

4.4.2 Definition. If the operators(Tt)t0 satisfy (a)TtfC0(Rd)for allfC0(Rd)and (b) limh0 Thf(x) =f(x) for allfC0(Rd), xRd, then(Tt) is called a Fellersemigroup.

4.4.3 Theorem. A Feller semigroup (Tt)t0 is a strongly continuous operator semi-group on C0(Rd), that is limh0 Thf = f holds in supremum norm. It is uniquelydetermined by its generatorA: D(A)C0(Rd)Rd with

Af := limh0

Thf fh

, D(A) :={fC0(Rd) | limh0

Thffh exists}.

Moreover, the semigroup uniquely defines the Markov kernels and thus the distributionof the associated Markov process (which is called Feller process).

4.4.4 Corollary. We have for allf D(A)ddt

Ttf=ATtf=TtAf.

4.4.5 Theorem. (Hille-Yosida) Let A be a closed linear operator on C0(Rd) withdense domainD(A). ThenA is the generator of a Feller semigroup if and only if

1. the range of0Id A is dense inC0(Rd) for some0 > 0;

2. if for some x Rd

and f D(A), f(x) 0 and f(x) = maxyRdf(x) thenAf(x)0 follows (positive Maximum principle).4.4.6 Theorem. Ifb and are bounded and satisfy the conditions of Theorem 4.3.1,then the Markov kernels (Px)xRd solving the martingale problem for (b, ) give riseto a Feller semigroup (Tt). Any functionfC20(Rd) lies inD(A) and fulfills

Af(x) =1

2

di,j=1

(T(x))ij2f

xixj(x) + b(x), grad(f)(x).


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4.4. The infinitesimal generator 43

We shall even prove a stronger result under less restrictive conditions, which turnsout to be a very powerful tool in calculating certain distributions for the solution

processes.4.4.7 Theorem. (Dynkins formula) Assume that b and are measurable, locallybounded and such that the SDE (2.1.1) with time-homogeneous coefficients has a (indistribution) unique weak solution. Then for allx Rd, fC2K(Rd) and all boundedstopping times we have

Ex[f(X())] =f(x) + Ex

0

Af(X(s)) ds

.

Proof. By Theorem 4.2.3 the process Mf is a local martingale under Px. By thecompact support off and the local boundedness ofb and we infer that Mf(t) is

uniformly bounded and therefore Mf is a martingale. Then the optional stoppingresultE[Mf()] = E[Mf(0)] = 0 yields Dynkins formula.

4.4.8 Example.

1. Let W be an m-dimensional Brownian motion starting in some point a andR := inf{t0 | W(t) R}. ThenEa[R] = (R2 a2)/m holds fora0. For this setR:= inf{t0 | X(t)R}, R > a,and consider a functionfC2(R) with compact support, f(0) = 0 and solvingAf(x) = 1 forx[0, R]. Then Dynkins formula yields

Ea[f(X(R n))] =f(a) + Ea[R n].For a similar functiong withAg = 0andg(0) = 0we obtainEa[g(X(Rn))] =g(a). Hence,

Ea[R n] = Ea[f(X(R n))] f(a)= Pa(X(R n) =R)f(R) + Ea[f(X(n))1{R>n}] f(a)= g(a) Ea[g(X(n))1{R>n}]f(R)g(R)+ Ea[f(X(n))1{R>n}] f(a)

follows. Using the uniform boundedness of f and g we infer by montone anddominated convergence forn

Ea[R] =g(a)f(R)

g(R) f(a).

Monotone convergence for R thus gives Ea[] < if and only iflimR

f(R)g(R)

is finite. The functions f and g can be determined in full gen-

erality, but we restrict ourselves to the case of vanishing drift b(x) = 0 andstrictly positive diffusion coefficientinf0yx (y)> 0 for allx >0. Then

f(x) =

x0

y0

2

2(z)dz dy andg(x) =x

will do. Sincef(x) , g(x) hold forx , LHopitals rule gives

limR

f(R)

g(R) = lim

R

f(R)

g(R) =

0

2

2(z)dz.

We conclude that the solution ofdX(t) =(X(t)) dW(t)withX(0) =a satisfies

Ea[] 0,Ea[] 1

2such that the rapid fluctuations ofX for

largex make excursions towards zero more likely.


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4.5. The Kolmogorov equations 45

4.5 The Kolmogorov equations

The main object one is usually interested in to calculate for the solution process Xof an SDE is the transition probability P(X(t)B | X(s) =x) forts0 and anyBorel set B. A concise description is possible, if a transition density p(x, y; t) existssatisfying

P(X(t)B | X(s) =x) =B

p(x, y; t s) dy.Here we shall present analytical tools to determine this transition density if it exists.The proof of its existence usually either relies completely on analytical results or onMalliavin calculus, both being beyond our scope.

4.5.1 Lemma. Assume thatb and are continuous and such that the SDE (2.1.1)

has a (in distribution) unique weak solution for any deterministic initial value. Forany f C2K(Rd) set u(x, t) := Ex[f(X(t))]. Then u is a solution of the parabolicpartial differential equation

u

t(x, t) = (Au(, t))(x), xRd, t0, withu(x, 0) =f(x) xRd .

Proof. Dynkins formula for =t yields by the Fubini-Tonelli theorem

u(x, t) =f(x) +

t0

Ex[Af(X(s))] ds xRd, t0.

Since the coefficients b and are continuous, the integrand is continuous and u iscontinuously differentiable with respect to t satisfying ut (x, t) = Ex[Af(X(t))]. Onthe other hand we obtain by the Markov property for t,h >0

Ex[u(X(h), t)] = Ex[EX(h)[f(X(t))]] = Ex[f(X(t + h))] =u(x, t + h).

For fixedt >0 we infer that the left hand side of

u(x, t + h) u(x, t)h

=Ex[u(X(h), t)] u(x, t)

h

converges forh0 to ut and therefore also the right-hand side. Therefore u lies inthe domain D(A) and the assertion follows.

4.5.2 Corollary. If the transition densityp(x, y; t) exists, is twice continuously dif-ferentiable with respect to x and continuously differentiable with respect to t, thenp(x, y; t) solves for ally Rd the backward Kolmogorov equation

u

t(x, t) = (Au(, t))(x), xRd, t0, withu(x, 0) =y(x).

In other words, for fixedy the transition density is the fundamental solution of thisparabolic partial differential equation.


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Proof. Writing the identity in the preceding lemma in terms ofp, we obtain for anyfC2K(Rd)

t Rd f(y)p(x, y; t) dy= ARd f(y)p(x, y; t) dy.By the compact support offand the smoothness properties ofp, we may interchangeintegration and differentiation on both sides. From

(tA)p(x, y; t)f(y)dy= 0 for

any test function fwe then conclude by a continuity argument.

4.5.3 Corollary. If the transition densityp(x, y; t) exists, is twice continuously dif-ferentiable with respect to y and continuously differentiable with respect to t, thenp(x, y; t) solves for allxRd the forward Kolmogorov equation

u

t(y, t) = (Au(, t))(y),

y R

d, t

0, withu(y, 0) =x

(y),

where

Af(y) =1

2

di,j=1

2

y2

(T(y))ijf(y)

di=1

yi

bi(y)f(y)

is the formal adjoint ofA. Hence, for fixedx the transition density is the fundamentalsolution of the parabolic partial differential equation with the adjoint operator.

Proof. Let us evaluate Ex[Af(X(t))] for anyfC2K(Rd) in two different ways. First,we obtain by definition

Ex[Af(X(t))] =

Af(y)p(x, y; t) dy=

f(y)(Ap(x, ; t))(y) dy.

On the other hand, by dominated convergence and by Dynkins formula we find f(y)

tp(x, y; t) dy=

tEx[f(X(t))] = Ex[Af(X(t))].

We conclude again by testing this identity with all f C2K(Rd).4.5.4 Remark. The preceding results are in a sense not very satisfactory because

we had to postulate properties of the unknown transition density in order to derive adetermining equation. Karatzas and Shreve (1991) state on page 368 sufficient condi-tions on the coefficientsb and, obtained from the analysis of the partial differentialequations, under which the transition density is the unique classical solution of the for-ward and backward Kolmogorov equation, respectively. Main hypotheses are ellipticityof the diffusion coefficient and boundedness of both coefficients together with certainHolder-continuity requirements. In the case of the forward equation in addition the

first two derivatives of and the first derivative of b have to have these properties,which is intuitively explained by the form of the adjointA.


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4.6. The Feynman-Kac formula 47

4.5.5 Example. We have seen that a solution of the scalar Ornstein-Uhlenbeckprocess

dX(t) =X(t) dt + dW(t), t0,is given byX(t) =X(0)et+

t0e(ts) dW(s). Hence, the transition density is given

by the normal density

p(x, y; t) = 1

22(2)1(e2t 1) exp(y xet)2/(21(e2t 1))

.

It can be easily checked thatp solves the Kolmogorov equations

u

t(x, t) =

2

2

2u

x2(x, t) +

u

x(x, t) and

u

t(y, t) =

2

2

2u

x2(y, t) u

y(y, t).

For = 0 and = 1 we obtain the Brownian motion transition densityp(x, y; t) =(2t)1/2 exp((yx)2/(2t)) which is the fundamental solution of the classical heatequation u

t = 1

22ux2

in both variablesx andy.

4.6 The Feynman-Kac formula


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Chapter 5

Stochastic control: an outlook

In this chapter we briefly present one main approach for solving optimal control prob-lems for dynamical systems described by stochastic differential equations: Bellmansprinciple of dynamic programming and the resulting Hamilton-Jacobi-Bellman equa-tion.

For some T > s0 and yRd we consider the controlled stochastic differentialequation

dX(t) =b(X(t), u(t), t) dt + (X(t), u(t), t) dW(t), t[s, T], X(s) =y,whereX isd-dimensional,W ism-dimensional Brownian motion and the coefficientsb : RdU [0, T] Rd, : RdU [0, T] Rdm are regular, say Lipschitzcontinuous, inx and depend on the controls u(t) taken in some abstract metric spaceU, which are Ft-adapted. The goal is to choose the control u in such a way that agiven cost functional

J(s, y; u) := E Ts

f(X(t), u(t), t) dt + h(X(t))

,

wheref andh are certain continuous functions, is minimized.

5.0.1 Example. A standard example of stochastic control is to select a portfolio ofassets, which is in some sense optimal. Suppose a riskless assetS0 like a bond growsby a constant rater >0 over time

dS0(t) =rS0(t) dt,

while a risky assetS1 like a stock follows the scalar diffusion equation of a geometricBrownian motion (Black-Scholes model)

dS1(t) =S1(t)

b dt + dW(t)

.

Since this second asset is risky, it is natural to supposeb > r. The agent has at eachtime t the possibility to trade, that is to decide the fraction u(t) of his wealthX(t)


52/56


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50 Chapter 5. Stochastic control: an outlook

Proof. See Theorem 4.3.3 in Yong and Zhou (1999).

Intuitively, this principle asserts that a globally optimal control u over the pe-riod [s, T] is also locally optimal for shorter periods [u, T]. In other words, we cannotimprove upon a globally optimal control by optimising separately on smaller subin-tervals. If this were the case, we could simply patch together these controls to obtaina globally better control.

The key point is that the knowledge of the value function for all arguments allowsto determine also the optimal controls which have to be applied in order to attain theoptimal cost. Therefore we have to study the equation for V in the Bellman principlemore thoroughly. Since integral equations are more difficult to handle, we look forinfinitesimal changes ins, which amounts to lettingzs appropriately. Heuristically,we interchange limit and infimum in the following formal(!) calculations, which have

to be justified much more accurately:

0 = 1

z s infuU(s,T)E zs

f(Xs,y,u(t), u(t), t) dt + V(z, Xs,y,u(z)) V(s, y)

then gives formally for zs

0 = inf uU(s,T)

f(y, u(s), s) +

tE[V(t, Xs,y,u(t))]|t=s)

,

which using the theory developed in the preceding chapter yields

0 = infuUf(y,u,s) + s V(s, y) + As,y,uV(s, y),where we have denoted by As,y,u the infinitesimal generator associated toXs,y,u:

As,y,uf(y) =1

2

di,j=1

(T(s,y,u))ij2f

yiyj(y) + b(s,y,u), grad(f)(y).

In terms of the so-called Hamiltonian

H(t,x,u,p,P) :=1

2trace(P(T)(s,y,u)) + b(s,y,u), p f(s,y,u)

we arrive at the Hamilton-Jacobi-Bellman (HJB) equation

V

s= sup

uUH

s,y,u, Vyi i, 2Vyiyj ij, (s, y)[0, T) Rd .Together with V(T, y) = h(y) we thus focus a terminal value problem for a partialdifferential equation.

In general, the value function only solves the HJB equation in a weak sense as aso called viscosity solution.


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51

In the sequel we assume that we have found the value function, e.g. via solvingthe HJB equation and proving uniqueness of the solution in a certain sense. Then the

optimal control u is given in feedback form u(t) = u

(X(t), t) with u

found by themaximizing property

H

s,y,u(y, s), Vyi

i, 2V

yiyj

ij

= supuUH

s,y,u, V

yi

i, 2V

yiyj

ij

.

For a correct mathematical statement we cite the standard classical verificationtheorem from (Yong and Zhou 1999, Thm. 5.5.1).

5.0.3 Theorem. SupposeW C1,2([0, T],Rd)solves the HJB equation together withits final value. Then

W(s, y)J(s, y; u)holds for all controlsu and all(s, y), that isWis a lower bound for the value function.Furthermore, an admissible controlu is optimal if and only if

V

t(t, Xs,y,u(t)) = H

t, Xs,y,u(t),u(t), V

yi(t, Xs,y,u(t))

i, 2V

yiyj(t, Xs,y,u(t))

ij

holds fort[s, T] almost surely.

Let us close this chapter by reconsidering the optimal investment example. TheHamiltonian in this case is given by

H(t,x,u,p,P) =1

22u2x2P+ (r+ (b r)u)xp

such that the HJB equation reads

tV(t, x) = supu[0,1]

1

22u2x2xxV(t, x) (r+ (b r)u)xxV(t, x)

.

Neglecting for a moment the restriction u[0, 1] we find the optimizing value u inthis equation by the first order condition

2ux2xxV(t, x) + (b r)xxV(t, x) = 0leading to the more explicit HJB equation

tV(t, x) =12

(r b)2

x2

(xV)2

2x2xx rxxV +(r b)

2

x2

(xV)2

2x2xxV

=rxxV +12

(r b)2x2(xV)22x2xx

.

Due to the good choice of the cost functional we find for (0, 1) a solution satisfyingthe HJB equation and having the correct final value to be

V(t, x) =e(Tt)x with = r+ (b r)222(1 ) .


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52 Chapter 5. Stochastic control: an outlook

This yields the optimal feedback function

u

(x, t) = b

r

2(1 ) .

Hence, ifu [0, 1] is valid, we have found the optimal strategy just to have a constantfraction of the wealth invested in both assets. Some special choices of the parametersmake the optimal choice clearer: for br we will not invest in the risky asset becauseit does not offer a higher average yield, for the same phenomenon occurs dueto the concavity of the utility function penalizing relative losses higher than gains,for0 or1 we do not run into high risk when investing in the stock and thuswill do so (even with borrowing for u >1!),.


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Bibliography

Durrett, R. (1996): Stochastic calculus. A practical introduction. Probability andStochastics Series. Boca Raton, FL: CRC Press.

Feller, W. (1971): An introduction to probability theory and its applications. Vol

II. 2nd ed.Wiley Series in Probability and Mathematical Statistics. New York etc.:John Wiley and Sons.

Kallenberg, O. (2002):Foundations of modern probability. 2nd ed.Probability andIts Applications. New York, Springer.

Karatzas, I., and S. E. Shreve(1991):Brownian motion and stochastic calculus.2nd ed. Springer, New York.

Kloeden, P. E., and E. Platen(1992):Numerical solution of stochastic differen-tial equations. Applications of Mathematics. 23. Berlin: Springer-Verlag.

Liptser, R. S., and A. N. Shiryaev (2001): Statistics of random processes. 1:General theory. 2nd ed. Springer, Berlin.

ksendal, B. (1998): Stochastic differential equations. An introduction with appli-cations. 5th ed. Universitext. Berlin: Springer.

Revuz, D., and M. Yor (1999): Continuous martingales and Brownian motion.3rd ed. Springer, Berlin.

Yong, J., and X. Y. Zhou (1999): Stochastic controls. Hamiltonian systems andHJB equations. Applications of Mathematics. 43. New York, NY: Springer.

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