3.3 Anticipating stochastic differential equations
3.3.2 Stochastic differential equations with boundary con-
Consider the following Stratonovich differential equation on the time inter- val [0,1], where instead of giving the valueX0, we impose a linear relation- ship between the values ofX0 andX1:
2 dXt=d
i=1Ai(Xt) ◦dWti+A0(Xt)dt,
h(X0, X1) = 0. (3.56)
We are interested in proving the existence and uniqueness of a solution for this type of equations. We will discuss two particular cases:
(a) The case where the coefficientsAi, 0≤i≤dand the functionhare affine (see Ocone and Pardoux [273]).
(b) The one-dimensional cased (see Donati-Martin [80]).
(A)Linear stochastic differential equations with boundary conditions Consider the following stochastic boundary value problem fort∈[0,1]
2 Xt=X0+d i=1
t
0AiXs ◦dWsi+t
0A0Xsds,
H0X0+H1X1=h. (3.57)
We assume that Ai, i = 0, . . . , d, H0, and H1 are m×m deterministic matrices andh∈Rm. We will also assume that them×2mmatrix (H0:H1) has rankm.
Concerning the boundary condition, two particular cases are interesting:
Two-point boundary-value problem: Letl ∈N be such that 0< l < m.
Suppose thatH0= H0′
0
,H1= 0
H1′′
, whereH0′ is anl×mmatrix and H1′′ is an (m−l)×m matrix. Condition rank(H0 :H1) =m implies that H0′ has rankl and thatH1′′has rank m−l. If we writeh=
h0
h1
, whereh0∈Rlandh1∈Rm−l, then the boundary condition becomes
H0′X0=h0, H1′X1=h1.
Periodic solution: SupposeH0=−H1 =Iand h= 0. Then the bound- ary condition becomes
X0=X1.
With (3.57) we can associate anm×madapted and continuous matrix- valued process Φ solution of the Stratonovich stochastic differential equa- tion
2 dΦt=d
i=1AiΦt◦dWti+BΦtdt,
Φ0=I. (3.58)
Using the Itˆo formula for the Stratonovich integral, one can obtain an ex- plicit expression for the solution to Eq. (3.57). By a solution we mean a con- tinuous and adapted stochastic processX such thatAi(Xt) is Stratonovich integrable with respect toWion any time interval [0, t] and such that (3.57) holds.
Theorem 3.3.4 Suppose that the random matrix H0+H1Φ1 is a.s. in- vertible. Then there exists a solution to the stochastic differential equation (3.57) which is unique among those continuous processes whose compo- nents belong to the spaceL2,4S,loc.
Proof: Define
Xt= ΦtX0, (3.59)
whereX0is given by
X0= [H0+H1Φ1]−1h. (3.60) Then it follows from this expression thatXi belongs to L2,4S,loc, for alli= 1, . . . , m, and due to the change-of-variables formula for the Stratonovich integral (Theorem 3.2.6), this process satisfies Eq. (3.57).
In order to show the uniqueness, we proceed as follows. Let Y ∈ L2,4S,loc(Rm) be a solution to (3.57). Then we have
Φ−t1=I− d i=1
t 0
Φ−s1Ai◦dWsi− t
0
Φ−s1A0ds.
By the change-of-variables formula for the Stratonovich integral (Theorem 3.2.6) we see that Φ−t1Yt, namely, Yt= ΦtY0.Therefore,Y satisfies (3.59).
But (3.60) follows from (3.59) and the boundary conditionH0Y0+H1Y1= h. Consequently,Y satisfies (3.59) and (3.60), and it must be equal toX.
Notice that, in general, the solution to (3.57) will not belong toL2,4S,loc(Rm).
One can also treat non-homogeneous equations of the form 2 Xt=X0+d
i=1
t
0AiXs ◦dWsi+t
0A0Xsds+Vt,
H0X0+H1X1=h, (3.61)
whereVtis a continuous semimartingale. In that case, Xt= Φt[H0+H1Φ1]−1
&
h−H1Φ1
1 0
Φ−s1◦dVs
' + Φt
t 0
Φ−s1◦dVs
is a solution to Eq. (3.61). The uniqueness in the class L2,4S,loc(Rm) can be established provided the processVtalso belongs to this space.
(B) One-dimensional stochastic differential equations with boundary conditions
Consider the one-dimensional stochastic boundary value problem 2 Xt=X0+ t
0σ(Xs)◦dWs+t
0b(Xs)ds,
a0X0+a1X1=a2, (3.62)
Applying the techniques of the anticipating stochastic calculus we can show the following result.
Theorem 3.3.5 Suppose that the functions σ and b1 := b+ 12σσ′ are of classC2with bounded derivatives anda0a1>0. Then there exists a solution to Eq. (3.62). Furthermore, if the functionsσ andb1 are of classC4 with bounded derivatives then the solution is unique in the class L2,4S,loc.
Proof: Letϕt(x) be the stochastic flow associated with the coefficientsσ and b1. By Theoren 3.3.1 for any random variableX0 the process ϕt(X0) satisfies
Xt=X0+ t
0
σ(Xs)◦dWs+ t
0
b(Xs)ds.
Hence, in order to show the existence of a solution it suffices to prove that there is a unique random variableX0such that
ϕ1(X0) =a2−a0X0
a1
. (3.63)
The mapping g(x) =ϕ1(x) is strictly increasing and this implies the exis- tence of a unique solution to Eq. (3.63).
Taking into account Theorem 3.3.2 to show the uniqueness it suffices to check that the unique solution X0 to Eq. (3.63) belongs to D1,ploc for some p >4. By the results of Doss (see [84] and Exercise 2.2.2) one can represent the flowϕt(x) as a Fr´echet differentiable function of the Brownian motion W. Using this fact and the implicit function theorem one deduces that
X0∈D1,ploc for allp≥2.
3.3.3 Stochastic differential equations in the Skorohod sense
LetW ={Wt, t∈[0,1]}be a one-dimensional Brownian motion defined on the canonical probability space (Ω,F, P). Consider the stochastic differen- tial equation
Xt=X0+ t
0
σ(s, Xs)dWs+ t
0
b(s, Xs)ds, (3.64) 0 ≤ t ≤ 1, where X0 is F1-measurable and σ and b are deterministic functions. First notice that the usual Picard iteration procedure cannot
be applied in that situation. In fact the estimation of the L2-norm of the Skorohod integral requires a bound for the derivative ofX, and this deriv- ative can be estimated only in terms of the second derivative. So we are faced with a nonclosed procedure. In some sense, Eq. (3.64) is an infinite dimensional hyperbolic partial differential equation. In fact, this equation can be formally written as
Xt = X0+ t
0
σ(s, Xs)◦dWs+1 2
t 0
σ′(s, Xs)! D+X
s+ D−X
s
"
ds +
t 0
b(s, Xs)ds.
Nevertheless, if the diffusion coefficient is linear it is possible to show that there is a unique global solution using the techniques developed by Buckdahn in [48, 49], based on the classical Girsanov transformation. In order to illustrate this approcah let consider the following particular case:
Xt=X0+σ t
0
XsdWs. (3.65)
WhenX0 is deterministic, the solution to this equation is the martingale Xt=X0eσWt−12σ2t.
IfX0= signW1, then a solution to Eq. (3.65) is (see Exercise 3.2.3) Xt= sign (W1−σt)eσWt−12σ2t.
More generally, by Theorem 3.3.6 below, if X0 ∈ Lp(Ω) for some p > 2, then
Xt=X0(At)eσWt−12σ2t
is a solution to Eq. (3.65), where At(ω)s=ωs−σ(t∧s). In terms of the Wick product (see [53]) one can write
Xt=X0 ♦eσWt−12σ2t.
Let us now turn to the case of a general linear diffusion coefficient and consider the equation
Xt=X0+ t
0
σsXsdWs+ t
0
b(s, Xs)ds, 0≤t≤1, (3.66) where σ∈L2([0,1]),X0 is a random variable and b is a random function satisfying the following condition:
(H.1) b: [0,1]×R×Ω→R is a measurable function such that there exist an integrable functionγt on [0,1], γt ≥0, a constant L >0 , and a set N1∈ F of probability one, verifying
|b(t, x, ω)−b(t, y, ω)| ≤ γt|x−y|, 1
0
γtdt≤L,
|b(t,0, ω)| ≤ L, for allx, y∈R, t∈[0,1] and ω∈N1.
Let us introduce some notation. Consider the family of transformations Tt, At: Ω→Ω, t∈[0,1] , given by
Tt(ω)s=ωs+ t∧s
0
σudu, At(ω)s=ωs−
t∧s 0
σudu . Note that TtAt=AtTt= Identity. Define
εt= exp t
0
σsdWs−1 2
t 0
σ2sds
.
Then, by Girsanov’s theorem (see Proposition 4.1.2) E[F(At)εt] = E[F] for any random variable F ∈L1(Ω). For eachx∈Rand ω∈Ω we denote byZt(ω, x) the solution of the integral equation
Zt(ω, x) =x+ t
0
ε−1s (Tt(ω)) b(s, εs(Tt(ω))Zs(ω, x), Ts(ω))ds . (3.67) Notice that for s ≤ t we have εs(Tt) = exps
0σudWu+12s 0 σ2udu
= εs(Ts). Henceforth we will omit the dependence on ω in order to simplify the notation.
Theorem 3.3.6 Fix an initial conditionX0∈Lp(Ω)for somep >2, and define
Xt=εtZt(At, X0(At)). (3.68) Then the process X = {Xt,0 ≤t ≤1} satisfies 1[0,t]σX ∈Domδ for all t ∈ [0,1], X ∈ L2([0,1]×Ω), and X is the unique solution of Eq. (3.66) verifying these conditions.
Proof:
Existence: Let us prove first that the processX given by (3.68) satisfies the desired conditions. By Gronwall’s lemma and using hypothesis (H.1), we have
|Xt| ≤εtetL
|X0(At)|+L t
0
ε−s1(Ts)ds
,
which implies supt∈[0,1]E(|Xt|q)<∞, for all 2≤q < p, as it follows easily from Girsanov’s theorem and H¨older’s inequality. Indeed, we have
E(|Xt|q) ≤ cqE
εqteqtL
|X0(At)|q+Lq t
0
ε−qs (Ts)ds
≤ cqeqL
E
εqt−1(Tt)|X0|q+Lqεqt−1(Tt) t
0
ε−sq(Ts2)ds
≤ C4
E(|X0|p)qp+ 15 .
Now fix t∈[0,1] and let us prove that 1[0,t]σX ∈Domδand that (3.66) holds. LetG∈ Sbe a smooth random variable. Using (3.68) and Girsanov’s theorem, we obtain
E t
0
σsXsDsGds
= E
t 0
σsεsZs(As, X0(As))DsGds
= E
t 0
σsZs(X0) (DsG) (Ts)ds
. (3.69) Notice that dsdG(Ts) =σs(DsG) (Ts). Therefore, integrating by parts in (3.69) and again applying Girsanov’s theorem yield
E t
0
Zs(X0)d
dsG(Ts)ds
= E
Zt(X0)G(Tt)−Z0(X0)G
− t
0
ε−s1(Tt)b(s, εs(Ts)Zs(X0), Ts)G(Ts)ds
=E(εtZt(At, X0(At))G)−E(Z0(X0)G)
− t
0
E(b(s, εsZs(As, X0(As)))G)ds
=E(XtG)−E(X0G)− t
0
E(b(s, Xs)G)ds . Because the random variable Xt−X0−t
0b(s, Xs)ds is square integrable, we deduce that 1[0,t]σX belongs to the domain of δ and that (3.66) holds.
Uniqueness: Let Y be a solution to Eq. (3.66) such thatY belongs to L2([0,1]×Ω) and1[0,t]σY ∈Domδ for all t∈[0,1]. Fix t∈[0,1] and let G be a smooth random variable. Multiplying both members of (3.66) by G(At) and taking expectations yield
E(YtG(At)) = E(Y0G(At)) +E t
0
b(s, Ys)G(At)ds
+E t
0
σsYsDs(G(At))
ds . (3.70)
Notice that dsdG(As) = −σ(DsG)(As). Therefore, integrating by parts obtains
E(YtG(At)) =E(Y0G)− t
0
E(Y0σs(DsG)(As))ds +E
t 0
b(s, Ys)G(As)ds
−E t
0
r 0
b(s, Ys)σr(DrG)(Ar)dsdr
+E t
0
σsYs(DsG)(As)ds
−E t
0
r 0
σsYs(DrDsG)(Ar)σrdsdr
. (3.71)
If we apply Eq. (3.71) to the smooth random variableσr(DrG)(Ar) for each fixed r∈[0, t], the negative terms in the above expression cancel out with the term Et
0σsYs(DsG) (As)ds
, and we obtain
E(YtG(At)) =E(Y0G) +E t
0
b(s, Ys)G(As)ds
. By Girsanov’s theorem this implies
E
Yt(Tt)ε−t1(Tt)G
=E(Y0G) +E t
0
b(s, Ys(Ts), Ts)ε−1s (Ts)G
ds.
Therefore, we have
Yt(Tt)ε−t1(Tt) =Y0+ t
0
b(s, Ys(Ts), Ts)ε−1(Ts)ds, (3.72) and from (3.72) we get Yt(Tt)ε−t1(Tt) =Zt(Y0) a.s. That is,
Yt=εtZt
At, Y0(At)
=Xt
a.s., which completes the proof of the uniqueness.
When the diffusion coefficient is not linear one can show that there exists a solution up to a random time.
Exercises
3.3.1 Letf be a continuously differentiable function with bounded deriv- ative. Solve the linear Skorohod stochastic differential equation
dXt = XtdWt, t∈[0,1]
X0 = f(W1).
3.3.2 Consider the stochastic boundary-value problem dXt1 = Xt2◦dWt,
dXt2 = 0,
X01 = 0, X11= 1.
Find the unique solution of this system, and show that E(|Xt|2) =∞for allt∈[0,1].
3.3.3 Find an explicit solution for the stochastic boundary-value problem dXt1 = (Xt1+Xt2)◦dWt1,
dXt2 = Xt2◦dWt2, X01+X02 = 1, X12= 1.
Notes and comments
[3.1] The Skorohod integral is an extension of the Itˆo integral introduced in Section 1.3 as the adjoint of the derivative operator. In Section 3.1, following [249], we show that it can be obtained as the limit of two types of modified Riemann sums, including the conditional expectation operator or subtracting a complementary term that converges to the trace of the derivative.
The forward stochastic integral, defined as δ(u) +
1 0
D−tutdt,
is also an extension of the Itˆo integral which has been studied by different authors. Berger and Mizel [30] introduced this integral in order to solve stochastic Volterra equations. In [14], Asch and Potthoff prove that it sat- isfies a change-of-variables formula analogous to that of the Itˆo calculus.
An approach using a convolution of the Brownian path with a rectangular function can be found in Russo and Vallois [298]. In [176] Kuo and Russek study the anticipating stochastic integrals in the framework of the white noise calculus.
The definition of the stochastic integral using an orthonormal basis of L2([0,1]) is due to Paley and Wiener in the case of deterministic integrands.
For random integrands this analytic approach has been studied by Balkan [16], Ogawa [274, 275, 276], Kuo and Russek [176] and Rosinski [293], among others.
[3.2] The stochastic calculus for the Skorohod and Stratonovich inte- grals was developed by Nualart and Pardoux [249]. In particular, the local
property introduced there has allowed us to extend the change-of-variables formula and to deal with processes that are only locally integrable or pos- sess locally integrable derivatives. Another extensive work on the stochastic calculus for the Skorohod integral inL2 is Sekiguchi and Shiota [305].
Other versions of the change-of-variables formula for the Skorohod inte- gral can be found in Sevljakov [306], Hitsuda [136], and ¨Ust¨unel [330].
[3.3] For a survey of this kind of applications, we refer the reader to Pardoux [278].
Stochastic differential equations in the Skorohod sense were first studied by Shiota [311] using Wiener chaos expansions. A different method was used by ¨Ust¨unel [331]. The approach described in Section 3.3, based on the Girsanov transformation, is due to Buckdahn and allows us to solve a wide class of quasilinear stochastic differential equations in the Skorohod sense with a constant or adapted diffusion coefficient. When the diffusion coeffi- cient is random, one can use the same ideas by applying the anticipating version of Girsanov’s theorem (see [50]). In [49] Buckdahn considers Skoro- hod stochastic differential equations of the form (3.64), where the diffusion coefficient σ is not necessarily linear. In this case the situation is much more complicated, and an existence theorem can be proved only in some random neighborhood of zero.
Stochastic differential equations in the sense of Stratonovich have been studied by Ocone and Pardoux in [272]. In this paper they prove the ex- istence of a unique solution to Eq. (3.49) assuming that the coefficient A0(s, x) is random. In [273] Ocone and Pardoux treat stochastic differen- tial equations of the Stratonovich type with boundary conditions of the form (3.56), assuming that the coefficientsAiand the functionhare affine, and they also investigate the Markov properties of the solution.
4
Transformations
of the Wiener measure
In this chapter we discuss different extensions of the classical Girsanov theorem to the case of a transformation of the Brownian motion induced by a nonadapted process. This generalized version of Girsanov’s theorem will be applied to study the Markov property of solutions to stochastic differential equations with boundary conditions.