evolution systems of cauchy-kowalewska and parabolic type with stochastic perturbations · 2010. 9....

EVOLUTION SYSTEMS OF CAUCHY-KOWALEWSKAAND PARABOLIC TYPE

WITH STOCHASTIC PERTURBATIONS

B. IFTIMIE and C. VÂRSAN

We consider nonlinear evolution systems with stochastic perturbations given byStratonovich stochastic integrals, for which we prove existence and uniquenessresults in a strong sense. The solution is locally defined up to an appropriatestopping time.

AMS 2000 Subject Classification: 35A10, 35K55, 60H15.

Key words: nonsingular representation, gradient system, Stratonovich integral,Cauchy-Kowalewska system, parabolic system.

1. INTRODUCTION

The investigation of evolution equations with stochastic perturbationsbecame mainly motivated by the increasing demands of large areas of appli-cability. Nowadays, this domain is of a large interest for physicians, biolo-gists, engineers, etc who are dealing with its applications. Concrete problemsarise in various fields as linear or nonlinear filtering (see [13], [8]), genetics,reaction-diffusion equations, random vibrations, neurophysiology, statisticalhydromechanics, front propagation in random media, pathwise stochastic con-trol theory, mathematical finance, etc.

Pardoux’s approach (see [12]) is based upon the theory of monotone op-erators while in [14] the main tool consists in the semigroup theory (evolutionoperators). Krylov [5] has a complete analytical approach to parabolic SPDEsand provides explicit formulas for the solutions of a certain class of SPDEs ofevolution type in [6].

Kunita [7] deals with second order stochastic linear parabolic differentialequations of the form

(1) du(t, x) = L(t)(u(t, x))dt +n∑

k=1

Mk(t)(u(t, x))dWk(t),

where L(t) is a second order elliptic (possibly degenerate) linear operatorand Mk(t) are first order ones. {W (t)} stands for a standard n-dimensionalBrownian motion and the stochastic integral is of Itô’s type.

MATH. REPORTS 10(60), 3 (2008), 213–238

214 B. Iftimie and C. Vârsan 2

The first order operator L(t) may be regarded as a random transportequation or wave equation in random media. In the case where L(t) is uni-formly elliptic, Pardoux [13] and Krylov–Rozovskii [5] proved the existenceand uniqueness of the solution (in a weak sense) by using analytic approaches.

Kunita also provides the existence of the solution for a slightly modi-fied equation of (1), where the stochastic integral is considered in the Fisk–Stratonovich sense. Consider

(2) u(t, x) = u0(x) +∫ t

0L(s)(u(s, x))ds +

n∑k=1

∫ t0

Mk(s)(u(s, x)) ◦ dWk(s),

for t ∈ [0, T ], where u0 ∈ L2(Rd). The method of construction of a solutionfor (2) is a method of perturbation which is widely used in analysis, by firstsolving the purely stochastic equation

v(t, x) = f +n∑

k=1

∫ t0

Mk(s)(v(s, x)) ◦ dWk(s).

and then applying a constants type variation formula.Krylov [6] studies more general systems than (1), namely,

(3) du(t, x) = L(t)(u(t, x))dA(t) +d∑

k=1

Mk(t)(u(t, x))dNk(t),

where L(t) and Mk(t) are second and first order elliptic operators, respectively,with possibly unbounded random coefficients, A(t) is a continuous increasingstochastic process and N(t) := (N1(t), . . . , Nd(t)) is a d-dimensional continu-ous local martingale.

As a remark, this type of equations was usually considered under theassumption of boundedness imposed on the coefficients. This is no longer thecase of evolution equations of a magnetic field evolving in a random media.

The solution belongs to some well-weighted Sobolev spaces. The tech-nique used here is to transform the initial system into deterministic PDEs withstochastic parameter, which also is the type of procedure employed by Kunita.

The evolution equations we are dealing with are not very different ofequations (2), i.e., L(t) will be a first or second order quasilinear operatorand, instead of Mk(t), we have diffusion vector fields which depend on thestate x and the unknown solution u(t, x), in a nonlinear way. The stochasticintegrals also are of Fisk–Stratonovich type. We prefer to deal with this type ofintegrals because they obey the rules of differentation of the ordinary calculusand one also can approximate the solution of a SDE written in Stratonovichform by the solutions of some ODEs.

The method of construction of the solution is based on decomposition.We consider first a purely differential stochastic system whose diffusion partis given by the original stochastic perturbation, for which we apply a resultfrom [16], concerning the nonsingular representation of the gradient system

3 Evolution systems of Cauchy-Kowalevska 215

associated with the diffusion vector field {gj , 1 ≤ j ≤ m}. This becomespossible by assuming that the Lie algebra generated by {gj , 1 ≤ j ≤ m} isfinite dimensional. We thus get an explicit formula for its solution via thecomposition of local flows generated by the vectors of a basis in L(g1, . . . , gm).

Next, a constant variation type formula reduces our problem to the res-olution of a classical system of PDEs with stochastic parameter for which weapply classical results concerning deterministic PDEs. The solution will be ameasurable and adapted process with respect to a given filtration.

The paper is divided into four sections. In Section 2 we construct asmooth approximation of a standard m-dimensional Brownian motion {W (t),0 ≤ t ≤ T}, which allows one to approximate the solution of a SDE withstochastic part written as a Fisk–Stratonovich integral, by the solutions of afamily of ODEs (by the so called Langevin’s procedure (see [9]), and we providea global nonsingular representation for the gradient system associated with asystem of C∞ vector functions generating a finite dimensional Lie algebra.

In Section 3 we deal with nonlinear evolution systems of Cauchy–Kowa-lewska type

du(t, x) =[ d∑

i=1

Ai(t, x, u(t, x))∂u

∂xi(t, x) + f(t, x, u(t, x))

]dt

+m∑

j=1

gj(x, u(t, x)) ◦ dWj(t),

u(0, x) = u0(x); t ∈ [0, T ], x ∈ V (0) ⊆ Rd, u ∈ RN .

We also provide an approximation of the solution by smooth and nonantici-pative solutions of some ODEs with stochastic parameter.

In the last section we study nonlinear SPDEs of parabolic type writtenin the Fisk-Stratonovich sense, described by

(4)

dui(t, x) = [4ui(t, x) + fi(t, x, u(t, x),∇xu(t, x))] dt,

+m∑

j=1

gij(x, u(t, x)) ◦ dWj(t), 1 ≤ i ≤ N,

u(0, x) = u0(x), t ∈ (0, T ], x ∈ Rn, u = (u1, . . . , uN ) ∈ RN .

The class of such SPDEs arises in a number of applications like filtering andpathwise stochastic control, mathematical finance. One can find in [10] acomplete list of the contributions related to the subject.

The general remarks in [10] still hold and the nonlinear drift is an ob-struction for the use of the classical martingale theory.

System (4) is analyzed using a finite dimensional Lie algebraic struc-ture generated by the corresponding diffusion vector fields {gj(x, u1, . . . , uN ),1 ≤ j ≤ m}.


2. BACKGROUND

Consider a standard m-dimensional Brownian motion {W (t), 0 ≤ t ≤ T}defined on a complete probability field {Ω,F , {Ft}0≤t≤T , P}. Define {vε(t),0 ≤ t ≤ T} by

vε(t) := W (t)−∫ t

0e−

1ε(t−s)dW (s), ε > 0.

It is obvious that {vε(t)} is an Ft-adapted process with C1 trajectories, becausethe integration by parts formula for semimartingales yields

vε(t) =1ε

∫ t0

W (s)e−1ε(t−s)ds.

The derivative of vε with respect to t is given by

dvεdt

(t) =1εW (t)− 1

ε2

∫ t0

W (s)e−1ε(t−s)ds =

1ε

∫ t0

e−1ε(t−s)dW (s).

By a direct computation we get

E |vε(t)−W (t)|2 ≤ ε for every t ∈ [0, T ].

Theorem 1. Assume that f(t, x), gj(t, x) : [0, T ] × Rn → Rn, j =1, . . . ,m, are bounded continuous functions such that

|h(t, x′′)− h(t, x′)| ≤ L|x′′ − x′|, t ∈ [0, T ], x′, x′′ ∈ Rn,

where L is a positive constant and h ∈ {f, g1, . . . , gm}. In addition, assumethat gj ∈ C1,2b ([0, T ]× R

n), i.e., gj ∈ C([0, T ]× Rn) is bounded and there exist∂gj∂t (t, x),

∂gj∂xk

(t, x), ∂2gj

∂xk∂xl(t, x), continuous and bounded mappings defined on

[0, T ]× Rn. Let x0(t) and xε(t), t ∈ [0, T ], be the solutions of

dx(t) =[f(t, x(t)) +

12

m∑j=1

∇xgj(t, x(t)) gj(t, x(t))]dt

+m∑

j=1

gj(t, x(t)) ◦ dWj(t), t ∈ [0, T ],

x(0) = x0 ∈ L2(Ω)

and dxεdt

(t) = f(t, xε(t)) +m∑

j=1

gj(t, xε(t))dvjεdt

(t), t ∈ [0, T ],

x(0) = x0.


The SDS may be rewritten as

x(t) = x0 +∫ t

0f(s, x(s))ds +

m∑j=1

∫ t0

gj(s, x(s)) ◦ dWj(s).

Thenxε(t) → x0(t) in L2(Ω) as ε → 0,

uniformly with respect to t ∈ [0, T ].Remark 1. Under the hypotheses of Theorem 1 we get the same type of

result if a stopping time is used. More precisely, we have

xε(t ∧ τ) → x0(t ∧ τ) in L2(Ω) as ε → 0,

uniformly with respect to t, where τ is an arbitrary stopping time of thefiltration Ft.

Let Λ be a finite dimensional Lie algebra and {Y1, . . . , YM} a basis in Λ(considered as a vectorial space).

Define the elements of Λ below:

(5)

X1 = Y1,X2(p2) = exp(t1 ad Y1) Y2,. . . . . . ,XM (pM ) = exp(t1 ad Y1) . . . exp(tM−1 ad YM−1)YM ,

where pj := (t1, . . . , tj−1), tj ∈ R, 2 ≤ j ≤ M .The next global nonsingular representation of a gradient system associ-

ated with the Xj(pj), 1 ≤ j ≤ M , holds.Theorem 2. Let {Y1, . . . , YM} and Xj(pj), 1 ≤ j ≤ M as above. Then

(6) (Y1, X2(p2), . . . , XM (t1, . . . , pM )) = (Y1, . . . , YM )×A(p),

where A(p) is a nonsingular (M × M)-matrix for every p ∈ RM such thatA(0) = IM and the components of A(p) are analytic functions. Moreover,

(7) (Y1, X2(t1), . . . , XM (t1, . . . , tM−1))× qi(p) = Yi, 1 ≤ i ≤ M,

where qi(p) ∈ Cω(RM ; RM ) is the ith column of the matrix (A(p))−1.Assume now Λ = C∞(D; Rn). Then it is not hard to see that the vector

fields in (5) may be rewritten as

(8)

X2(t1; y) = H1(−t1, y1) Y2(G1(−t1, y1)),. . . . . .Xm(t1, . . . , tm−1, y) = H1(−t1, y1) H2(−t2, y2) . . .×

×Hm−1(−tm−1, ym−1) Ym(ym),


where y ∈ V (x0), p = (t1, . . . , tm) ∈ Dm and

Hi(t, y) := (∇xGi(t, y))−1 , yi+1 = Gi(−ti, yi), y1 = y,ti ∈ (−ai, ai), i = 1, . . . ,m− 1.

and Gi(ti, x) stands for the local flow defined by the vector field Yi(y). Set also

(9) G(p, x) := G1(t1) ◦ . . . ◦Gm(tm)(x),

where p = (t1, . . . , tm) ∈ Dm :=m∏

i=1(−ai, ai) and x ∈ V (0) ⊆ Rn.

Theorem 3. Let {Y1, . . . , Ym} ⊆ C∞(D; Rn) such that the Lie algebraΛ = L(Y1, . . . , Ym) generated by the linear independent vectors Y1, . . . , Ym isfinite dimensional. If {Y1, . . . , Ym, Ym+1, . . . , YM} is a basis in Λ, then thereexist q1, . . . , qm ∈ Cω(DM ; RM ) such that(10) ∇pG(p, x)× qj(p) = Yj(G(p, x)), (∀) p ∈ DM , x ∈ V (x0), 1 ≤ j ≤ m.Here, G is the composition of the local flows G1, ..., GM generated by Y1, ..., YM .

Notice that the vectors Y1, . . . , Ym (which generate the Lie algebra Λ)may be recovered along the orbit G(p, x), starting with the flow itself and themappings qi(p).

Remark 2. The hypothesis in Theorem 3 which states that Λ = L(Y1, . . . ,Ym) is a finite dimensional Lie algebra is fulfilled if, for instance, Yi(y) =ai + 〈bi, y〉, i = 1, . . . ,m, with ai, bi ∈ Rn. It also holds under the assumptionthat Λ is a nilpotent algebra, i.e., the (M×M) matrix Bj of the linear operatorad Yj with respect to the basis {Y1, . . . , YM} is a nilpotent one (B

Njj = 0M , 1 ≤

j ≤ M , for some natural number Nj).

3. SPDEs OF CAUCHY–KOWALEWSKA TYPE

We treat here nonlinear differential systems of the form

(11)

du(t, x) =

[d∑

i=1

Ai(t, x, u(t, x))∂u

∂xi(t, x) + f(t, x, u(t, x))

]dt

+m∑

j=1

gj(x, u(t, x)) ◦ dWj(t),

u(0, x) = u0(x), t ∈ [0, T ], x ∈d∏

j=1

{xj ∈ R | |xj | ≤ R}, u ∈ RN .

{W (t), t ∈ [0, T ]} stands for a standard m-dimensional Brownian motion overa filtered probability space {Ω,F , (Ft)t∈[0,T ], P}, and the filtration (Ft) satisfiesthe usual conditions.


For 0 ≤ s < 1 set Ds :=d∏

j=1{xj ∈ C | |xj | ≤ sR}, Es :=

N∏i=1{ui ∈ C |

|ui| ≤ sγ}, D′s :=d∏

j=1{xj ∈ R | |xj | ≤ sR}, E′s :=

N∏i=1{ui ∈ R | |ui| ≤ sγ}.

Let Xs be the Banach space of N -dimensional functions which are holo-morphic and bounded in Ds, endowed with the norm

‖w‖s := supx∈Ds

|w(x)|.

The family {Xs | 0 ≤ s < 1} forms a scale of Banach spaces.In this section we assume that the following hypotheses, denoted (H),

are fulfilled.(i1) The components of Ai(t, x, u) and f(t, x, u) are continuous functions

of t ∈ [0, T ] with values in the Banach space of bounded and holomorphicfunctions in a neighbourhood of D1 × E1, such that their restrictions to theset [0, T ]×D′1 × E′1 are real-valued (analytic) functions.

(i2) Ai(t, x, u) and f(t, x, u) together with their partial derivatives of thefirst two orders with respect to xj and ui are bounded by a positive constantC for any t ∈ [0, T ], x ∈ D1, u ∈ E1.

(i3) The components of u0 are analytic functions on D′1.(i4) The components of gj are analytic functions on D′1 × E′1 for any

1 ≤ j ≤ m.(i5) For any x ∈ D′1, the Lie algebra Λ(x) generated by the fields gj(x, ·),

1 ≤ j ≤ m, with respect to the usual Lie product, is finite dimensional uni-formly with respect to x, i.e., there exists a system of generators

{Y1(x, ·), . . . , YM (x, ·)} ⊆ Λ(x)

such that for any Y (x, ·) ∈ Λ(x) we may write Y (x, u) =∑M

k=1 αkYk(x, u) forany x ∈ D′1, u ∈ E′1. The coefficients αk do not depend on x, u but only onYj and Y .

Remark 3. Hypothesis (i5) is fulfilled if the vectors gj(x, ·), 1 ≤ j ≤ m, arelinear, such that gj(x, u) = 〈aj(x), u〉 + bj(x), with the functions aj(x), bj(x)having a “nice” behaviour with respect to the variable x. It is also true if Λ(x)is a nilpotent algebra, uniformly with respect to x ∈ D′1.

Remark 4. This assumption is of hypoellipticity type. Kunita [7, pp.148–153] also uses this type of hypothesis, when the hypoellipticity of someZakai equation is discussed.

In [8] is studied the SDE

(12) dξt =r∑

j=1

Xj(ξt) ◦ dM jt ,


where Xj , 1 ≤ j ≤ r, are C∞ vector fields on a σ-compact connected C∞-manifold M and M jt , 1 ≤ j ≤ r, are continuous semimartingales. Assumingthat the Lie algebra generated by the complete vector fields X1, . . . , Xr is finitedimensional, it is shown that the solution ξt of equation (12) is conservative anda C∞ diffeomorphism of the manifold M for any t > 0, a.e. ω (see Theorem 3.8,page 236, of the same reference). Moreover, considering a basis {Y1, . . . , YM}of L(X1, . . . , Xr), the solution of (12) may be written under an explicit form(see Theorem 6.1, page 250). In this case we have a decomposition result forthe solution ξt.

Hypothesis (i5) is used in order to get an explicit formula for the solutionof the SDEs

dy(t) =m∑

j=1

gj(x, y(t)) ◦ dWj(t).

We now give a rigorous meaning to the solution of system (11).

Definition 1. Let s0 ∈ [0, 1) be fixed. We say that the pair (u, τ) consistingof a process u(t, x, ω) : [0, r0) × D′s0 × Ω → E

′1 and a stopping time τ of the

filtration {Ft}, is a solution of system (11) if(a) u(· , x) is an adapted continuous process for any x ∈ D′s0 .(b) the components of u(t, · , ω) are analytic functions for any t ∈ [0, r0)

a.e. ω.(c) u(t, x) fulfills

u(t, x) = u0(x) +∫ t

0

[d∑

i=1

Ai(s, x, u(s, x))∂u

∂xi(s, x)+f(s, x, u(s, x))

]ds(13)

+m∑

j=1

∫ t∧τ0

gj(x, u(s, x)) ◦ dWj(s), ∀t ∈ [0, r0), x ∈ D′s0 .

The stochastic integrals are understood in the Fisk–Stratonovich sense.Denote by Gi(t, x, λ), t ∈ [−ai, ai], x ∈ [−R,R]d, λ ∈ [−γi, γi]N , 1 ≤ i ≤

M , the local flow generated by Yi(x, ·), i.e., Gi solves∂Gi∂t

(t, x, λ) = Yi(x, Gi(t, x, λ)),

Gi(0, x, λ) = λ,

where x is a parameter.

Remark 5. The positive numbers ai, γi are independent of x ∈ D′1(by (i5)) and such that Gi([−ai, ai] × D′1 × [−γi, γi]N ) ⊆ [−γi−1, γi−1]N , fori = 1, . . . ,M and γ0 := γ.

By classical results concerning the behaviour of the solutions of ordi-nary differential equations with respect to the parameters, we have Gi ∈Cω([−ai, ai]×D′1 × [−γi, γi]N ), 1 ≤ i ≤ M .


Define also the mapping G(p, x, λ) as the composition of local flowsGi(t, x, λ), 1 ≤ i ≤ M , i.e.,

G(p, x, λ) := G1(t1, x) ◦G2(t2, x) ◦ . . . ◦GM (tM , x)(λ),

where p = (t1, t2, . . . , tM ) ∈ IM :=M∏i=1

[−ai, ai], x ∈ D′1, λ ∈ U := [−ρ, ρ]N and

ρ := min{γ1, . . . , γM}.G(p, x, λ) ∈ Cω(IM × D′1 × U) and is an analytic diffeomorphism with

respect to λ ∈ U .The analysis of the gradient system associated with a family of vector

fields which generate a finite dimensional Lie algebra, mainly its nonsingularrepresentation (see Section 1), yields the existence of some analytic vectorfunctions qj(p), p ∈ IM , 1 ≤ j ≤ m, such that(14) ∇pG(p, x, λ) qj(p) = gj(x,G(p, x, λ))for any (p, x, λ) ∈ IM ×D′1 × U , 1 ≤ j ≤ m.

Consider the stochastic differential system

(15) dy(t) =m∑

j=1

gj(x, y(t)) ◦ dWj(t), y(0) = λ ∈ U.

We first solve the auxiliary SDE

(16) p(t) =m∑

j=1

∫ t0

qj(p(s)) ◦ dWj(s), p(0) = 0.

The diffusion fields of the last system are not globally defined and do notobey the usual Lipschitz and linear growth conditions. In order to solve it,we define a C∞0 function ϕ(p) whose support is contained in the closed ballBδ := {p ∈ Rm | ‖p‖ ≤ δ}, where 0 < δ ≤ 12 min{a1, . . . , aM}.

Set q̃j(p) := ϕ(p)qj(p). The SDS

(17) p(t) =m∑

j=1

∫ t0

q̃j(p(s)) ◦ dWj(s)

satisfies the conditions of existence and uniqueness of the solution. Let p̃ (t)be its solution.

Define now the stopping time τ as the first exit time of the process p̃ (t)from the closed ball Bδ, that is,

(18) τ := inf{t ∈ [0, T ] | ‖p̃ (t)‖ ≥ δ}.It follows that the stopped process p̂ (t) := p̃ (t ∧ τ) takes values in Bδ andsatisfies

(19) p̂(t) =m∑

j=1

∫ t∧τ0

qj(p̂ (s)) ◦ dWj(s) =m∑

j=1

∫ t0

χ[0,τ ](s)qj(p̂ (s)) ◦ dWj(s).


In this way we obtain a solution of system (15), but only up to the stoppingtime τ .

Set now v̂ (t, x, λ) := G(p̂ (t), x, λ). Applying Itô’s stochastic calculusrules and taking into account formulas (14) and (19), we easily get

v̂ (t, x, λ) = λ +m∑

j=1

∫ t0

χ[0,τ ](s) ∇ugj(x, v̂ (s, x, λ)) gj(x, v̂ (s, x, λ))ds

+m∑

j=1

∫ t0

χ[0,τ ](s) gj(x, v̂ (s, x, λ))dWj(s)

= λ +m∑

j=1

∫ t0

χ[0,τ ](s) gj(x, v̂ (s, x, λ)) ◦ dWj(s)

= λ +m∑

j=1

∫ t∧τ0

gj(x, v̂ (s, x, λ)) ◦ dWj(s)

for t ∈ [0, T ], x ∈ D′1, λ ∈ U . Notice that v̂ is well defined by the choice of τ .The process v̂ (t, x, λ) is adapted, has continuous trajectories and its

components are analytic functions of (x, λ).In order to find a solution of system (11), we apply a constant variation

formula, i.e., we are looking for u(t, x) of the form

u(t, x) := v̂ (t, x, λ(t, x)) = G(p̂ (t), x, λ(t, x)),

with λ(t, x) solving some PDEs with stochastic parameter for which we applyan abstract Cauchy–Kowalewska theorem. The domain of the solution λω(t, x)will not depend on the stochastic parameter ω.

Remark 6. The mappings qj are independent of x, by (i5) in hypothesis(H). Thus, the stopping time τ does not depend on x and the differentiationof v̂ with respect to the state variable x does not create any difficulty.

Consider now the partial differential system with stochastic parameter

(20)

∂λ

∂t(t, x) =

d∑i=1

Āi(t, x, λ(t, x))∂λ

∂xi(t, x) + f̄(t, x, λ(t, x)),

λ(0, x) = u0(x); t ∈ [0, T ], x ∈ D1,where

Āi(t, x, λ) := [∇λG(p̂ (t), x, λ)]−1 Ai(t, x, G(p̂ (t), x, λ)) ∇λG(p̂ (t), x, λ),

(21) f̄(t, x, λ) := [∇λG(p̂ (t), x, λ)]−1[f(t, x, G(p̂ (t), x, λ))

+d∑

i=1

Ai(t, x, G(p̂ (t), x, λ))∂G

∂xi(p̂ (t), x, λ)

].


Remark 7. The dependence of these coefficients on the random parameteris given through the uniformly bounded process p̂ (t). Thus, the mappingv̂ (t, x, λ), defined on [0, T ]×D1×U, together with its partial derivatives withrespect to xi, λj are uniformly bounded and the same holds for its smoothinverse with respect to λ.

Taking now into account hyphotheses (H) and the properties of the ran-dom field G(p̂ (t), x, λ), it is not hard to see that the components of the coef-ficients of system (20) are continuous in t, with values which are holomorphicfunctions in a neighborhood of D1×U . In addition, their first and second par-tial derivatives with respect to xj and λi are bounded by some deterministicconstant L. We require that the solution λ(t, x) takes values in U , in order todefine properly the mappings Āi(t, x, λ(t, x)), f̄(t, x, λ(t, x)).

Let ω ∈ Ω be fixed and consider the system

(22)

dudt

(t) = Fω(u(t), t), t ∈ [0, T ],

u(0) = u0,

where Fω : {u ∈ X1 | ‖u‖1 < ρ} × [0, T ] → X1 is given by

Fω(u, t)(·) :=d∑

i=1

Āi(t, · , u(·), ω)∂u

∂xi(·) + f̄(t, ·, u(·), ω).

Define the linear operator Au(t) on Xs and taking values in any Xs′ , 0 ≤ s′ < s,as

Au(t)(v)(x) :=d∑

i=1

Āi(t, x, u(x))∂v

∂xi(x)+

N∑k=1

d∑i=1

∂Āi∂λk

(t, x, u(x))∂u

∂xi(x) vk(x) +∇λf̄(t, x, u(x)) v(x), v ∈ Xs, x ∈ Ds′ .

It can be shown by standard computations that system (22) obeys hypotheses(1.1)–(1.6) of Theorem 1.1 in [11]. Thus we are in a position to state

Proposition 1. Let a be the limit of the sequence (ak) in formula (2.7)of [11]. For any s0 ∈ (0, 1) there exists a mapping λ(t, x, ω) : [0, a(1 − s0)) ×D′s0 × Ω → R

N in C1(Is0 ;Xs0) such that(a) the components of λ(t, · , ω) are analytic functions for any fixed (t, ω) ∈

Is0 × Ω;(b) λ solves system (20);(c) ‖λ(t, x, ω)‖ < ρ, (∀) t ∈ Is0, x ∈ D′s0, a.e. ω ∈ Ω.Moreover, if λ̃ is a solution of (20) such that λ̃ ∈ C1(([0, a(1−s));Xs) for

any s ∈ (0, 1), then λ(t, x) and λ̃(t, x) are indistinguishable processes. Here,Is0 := [0, a(1− s0)).


Remark 8. λ(· , x, ω) is an adapted process with C1 trajectories for anyx ∈ D′s0 . This is easily seen by considering the iterative procedure of con-struction of the solution and an induction argument (for further details, see[11, pages 566, 567]).

In what follows, let s0 ∈ (0, 1) be fixed and define

(23) u(t, x) := G(p̂ (t), x, λ(t, x)) = v̂ (t, x, λ(t, x)),

for (t, x) ∈ Is0 ×Ds0 . Recall that the stopping time τ was defined in (18).The main result of this section is given by

Theorem 4. The pair (u, τ) is a solution of system (11) in the sense ofDefinition 1 such that u ∈ C1(Is0 ;Xs0) for every s0 ∈ (0, 1). In addition, let(u1, τ), (u2, τ) be solutions of (13) satisfying |u1(t, x)| ≤ γ, |u1(t, x)| ≤ γ forany t ∈ [0, a(1− s0)) and x ∈ Ds0 , and u1, u2 ∈ C1([0, a(1− s0));Xs0) for anys0 ∈ (0, 1). Then u1(t, x, ω) = u2(t, x, ω) for any t ∈ [0, a(1− s0) and x ∈ Ds0 ,a.e. ω.

Proof. Notice first that the process u(t, x) in (23) is well defined bystatement (c) in Proposition 1 and the way τ is defined.

Next, on account of the analiticity of the components of G(t, · , ·) andλ(t, ·), the components of the mapping u(t, ·) also are analytic for fixed t, ω.The fact that u(· , x) is a continuous adapted process for a fixed x relies on thesame property satisfied by the processes p̂ (t), λ(t, x) and on the smoothnessof G.

Define now the (M + n)-dimensional process

Xt(x) :=

(p̂ (t)

λ(t, x)

).

Extending the locally defined field G to a C1,2,2(RM ×Rd×RN ; RN ) mapping,Itô’s formula yields

(24) u(t, x) = G(Xt(x), x) = u(0, x) +∫ t

0∇pG(p̂ (s), x, λ(s, x))dp̂ (s)

+∫ t

0∇λG(p̂ (s), x, λ(s, x))dλ(s, x)

+M∑

i,j=1

m∑k=1

12

∫ t0

∂2G

∂ti∂tj(p̂ (s), x, λ(s, x))qik(p̂ (s))q

jk(p̂ (s))ds.

We used the fact that the trajectories of the process λ(· , x) have boundedvariation and p̂ is a continuous Brownian martingale, which means that thecross variation process 〈p̂ (·), λ(· , x)〉t is identically 0.


We can write

u(t, x) +∫ t

0∇λG(p̂ (s), x, λ(s, x))dλ(s, x)

= u(0, x) +d∑

i=1

∫ t0

Ai(s, x,G(p̂ (s), x, λ(s, x)))

×[∇λG(p̂ (s), x, λ(s, x))

∂λ

∂xi(s, x) +

∂G

∂xi(p̂ (s), x, λ(s, x))

]ds

+∫ t

0f(s, x,G(p̂ (s), x, λ(s, x)))ds

= u(0, x) +∫ t

0

[ d∑i=1


∂xi(s, x) + f(s, x, u(s, x))

]ds.

We took into account the system satisfied by λ(t, x) and the formulas for thefirst order partial derivatives of u with respect to xi, namely,∂u

∂xi(t, x) =

∂G

∂xi(p̂ (t), x, λ(t, x)) +∇λG(p̂ (t), x, λ(t, x))

∂λ

∂xi(t, x), 1 ≤ i ≤ d.

The sum of the remaining terms on the r.h.s. of (24) is equal to

12

M∑i=1

∫ t∧τ0

∂G

∂p(p̂ (s), x, λ(s, x)) ∇pqi(p̂ (s)) qi(p̂ (s))ds

+12

M∑i,j=1

m∑k=1

∫ t∧τ0

∂2G

∂ti∂tj(p̂ (s), x, λ(s, x)) qik(p̂ (s)) q

jk(p̂ (s))ds

+M∑i=1

∫ t∧τ0

∇pG(p̂ (s), x, λ(s, x)) qi(p̂ (s))dWi(s)

=M∑i=1

12

∫ t∧τ0

qi(p̂ (s)) ∇p(∇pG(· , x, λ(s, x))qi(·)

)(p̂ (s))ds

+m∑

i=1

∫ t∧τ0

gi(G(p̂ (s), x, λ(s, x)))dWi(s).

We also have

qi(p̂ (s))∇p(∇pG(· , x, λ(s, x)) qi(·)

)(p̂ (s))

= qi(p̂ (s)) ∇p(gi(G(· , x, λ(s, x)))

)(p̂ (s))

= ∇ugi(G(p̂ (s), x, λ(s, x))) ∇pG(p̂ (s), x, λ(s, x)) qi(p̂ (s))= ∇ugi(G(p̂ (s), x, λ(s, x))) gi(G(p̂ (s), x, λ(s, x))).


Finally, considering all previous formulas and the initial condition

u(0, x) = G(0, x, λ(0, x)) = λ(0, x) = u0(x),

we get

u(t, x) = u0(x) +∫ t

0

[ d∑i=1


∂xi(s, x) + f(s, x, u(s, x))

]ds

+∫ t∧τ

0gj(x, u(s, x)) ◦ dWj(s).

Hypothesis (H) and the results stated in Proposition 1 imply that u ∈C1([0, a(1− s0));Xs0) for every s0 ∈ (0, 1).

The diffeomorphism property of the orbit G with respect to the variableλ leads us to the local uniqueness result.

Let H(p, x, u), defined on IM ×D1×U , be the inverse of G with respectto λ, i.e.,

H(p, x, u) := GM (−tM , x) ◦ . . . ◦G1(−t1, x)(u),

for p = (t1, . . . , tM ) ∈ IM , x ∈ D1, u ∈ U . Hence the systems uk(t, x) =G(p̂ (t), x, λk), 1 ≤ k ≤ 2, admit unique solutions, given by λk(t, x) = H(p̂ (t),x, uk(t, x)). It is not hard to see that for any 0 < s0 < 1, λk ∈ C1(([0, a(1 −s0));Xs0). Indeed, choose t1, t2 ∈ [0, a(1− s0). Since

|λk(t2, x)− λk(t1, x)| = |H(p̂ (t2), x, uk(t2, x))−H(p̂ (t1), x, uk(t1, x))|

≤ |p(t2)− p(t1)|∫ 1

0|∇pH(αp(t1) + (1− α)p(t2), x, uk(t2, x))|dα

+|u2(t, x)− u1(t, x)|∫ 1

0|∇uH(p(t1), x, αu1(t, x) + (1− α)u2(t, x))|dα,

using the fact that H together with its first partial derivatives with respect top and u are bounded, we get

‖λk(t2)− λk(t1)‖s0 ≤ L(|p(t2)− p(t1)|+ ‖uk(t2)− uk(t1)‖s0).

Taking also into account that uk ∈ C1([0, a(1 − s0));Xs0), we deduce thatλk ∈ C1([0, a(1− s0));Xs0) for a.e. ω ∈ Ω.

We next apply Itô’s rules to the process G(Y kt (x), x), where

Y kt (x) :=(

p̂(t)λk(t, x)

).


A straightforward computation yields

uk(t, x) = G(Y kt (x), x) = u0(x) +∫ t

0∇pG(p̂ (s), x, λk(s, x))dp̂ (s)

+∫ t

0∇λG(p̂ (s), x, λk(s, x))

∂λk∂t

(s, x)ds

+M∑

i,j=1

m∑l=1

12

∫ t0

∂2G

∂ti∂tj(p̂ (s), x, λk(s, x)) qil(s) q

jl (s)ds

=∫ t

0∇pG(p̂ (s), x, λk(s, x))

∂λk∂t

(s, x)ds +m∑

i=1

∫ t∧τ0

gi(x, uk(s, x)) ◦ dWi(s).

We also used the formula

λk(0, x) = G(p̂(0), x, λk(0, x)) = uk(0, x) = u0(x).

Recall that (uk, τ), k = 1, 2, are solutions of (13) and comparing with theprevious expressions we arrive at the following PDEs with random parametersatisfied by λk:

(25)

∇λG(p̂ (t), x, λk(t, x))

∂λk∂t

(t, x) = Ai(t, x, uk(t, x))∂uk∂xi

(t, x)

+f(t, x, uk(t, x)),

λk(0, x) = u0(x).

The first order partial derivatives of uk with respect to xi are given by∂uk∂xi

(t, x) =∂G

∂xi(p̂ (t), x, λk(t, x)) +∇λG(p̂ (t), x, λk(t, x))

∂λk∂xi

(t, x).

Inserting now the last formula in system (25) it is easy to see that λk satis-fies (20). We use Proposition 1 to deduce that the processes λ1 and λ2 areindistinguishable, and the same will hold for u1 and u2. �

3.1. APPROXIMATION OF THE SOLUTIONBY NONANTICIPATIVE SMOOTH PROCESSES

We now show that the solution of system (11) may be approximated inL2(Ω) by the nonanticipative solutions of some ordinary differential systemswith random parameter. This is accomplished by using approximations ofLangevin’s type (see Subsection 1.1).

Recall that we defined

vε(t) :=1ε

∫ t0

e−1ε(t−s)W (s)ds

for ε > 0. Denote αjε(t) = dvjε

dt (t).


The results stated in Section 1 allow us to approximate the solution p̂ (t),t ∈ [0, T ], in L2(Ω) by the solutions p̂ε(t) of the ODEs

(26)

dp̂εdt

(t) =m∑

j=1

χ[0,τ ](t) qj(p̂ε(t))αjε(t),

p̂ε(0) = 0,

i.e.,

(27) limε→0

E[‖p̂ε(t)− p̂ (t)‖2

]= 0 uniformly with respect to t ∈ [0, T ].

Fix s0 ∈ (0, 1) and denote r0 := a(1− s0). We know that we may defineus0 : [0, r0)×Ds0 as the solution of system (11) such that us0 ∈ C1([0, r0);Xs0).Choose an arbitrary s1 ∈ (0, s0). Set(28) uε(t, x) := G(p̂ε(t), x, λ(t, x))

for t ∈ [0, a(1 − s0)), x ∈ Ds1 . Clearly, uε is differentiable with respect to tand the partial derivative with respect to t is given by

(29)∂uε∂t

(t, x)=∇pG(p̂ε(t), x, λ(t, x))dp̂εdt

(t)+∇λG(p̂ε(t), x, λ(t, x))∂λ

∂t(t, x).

Recall that λ(t, x) is the solution of∂λ

∂t(t, x) =

d∑i=1

Āi(t, x, λ(t, x))∂λ

∂xi(t, x) + f̄(t, x, λ(t, x))

λ(0, x) = u0(x); t ∈ [0, T ], x ∈ D1.

Integrating now formula (29) yields

(30) uε(t, x) = uε(0, x) +m∑

j=1

∫ t∧τ0

∇pG(p̂ε(s), x, λ(s, x)) qj(p̂ε(s)) αjε(s)ds

+∫ t

0∇λG(p̂ε(s), x, λ(s, x))

[ d∑i=1

(∇λG)−1(p̂ (s), x, λ(s, x)

)×Ai(s, x, u(s, x)) ∇λG(p̂ (s), x, λ(s, x))

∂λ

∂xi(s, x)

]ds

+∫ t

0∇λG(p̂ε(s), x, λ(s, x)) (∇λG)−1

(p̂ (s), x, λ(s, x)

)×[f(s, x, u(s, x)) +

d∑i=1

Ai(s, x, u(s, x))∂G

∂xi

(p̂ (s), x, λ(s, x)

)]ds

:= u0(x) + I1ε (t, x) + I2ε (t, x) + I

3ε (t, x),


where

uε(0, x) = G(p̂ (0), λ(0, x)) = λ(0, x) = u0(x), (∀) x ∈ Ds1 , a.e. ω.

The integral terms I2ε (t, x) and I3ε (t, x) are of the type

Jε(t, x) =∫ t

0∇λG(p̂ε(s), x, λ(s, x)) h(s, x)ds,

where the random mapping h is continuous and bounded by a nonrandomconstant.

We next prove

(31) Jε(t, x)→J(t, x) :=∫ t

0∇λG(p̂ (s), x, λ(s, x))h(s, x)ds in L2(Ω) as ε→0,

uniformly with respect to t, x. We have

E(‖Jε(t, x)− J(t, x)‖

)2≤ E

[ ∫ t0‖∇λG(p̂ε(s), x, λ(s, x))

−∇λG(p̂ (s), x, λ(s, x))‖ |h(s, x)|ds]2

≤ C supp∈Bδ(0)x∈Ds0

λ∈Bρ(0)

∥∥∥ ∂2G∂λk∂tj

(p, x, λ)∥∥∥2 E [∫ t

0(‖p̂ε(s)− p̂ (s)‖)ds

]2

≤ CLT∫ t

0E[‖p̂ε(s)− p̂ (s)‖2

]ds.

By (27), we deduce that (31) holds. Hence

I2ε (t, x) + I3ε (t, x) →

∫ t0

[ d∑i=1


∂xi(s, x) + f(s, x, u(s, x))

]ds

in L2(Ω) as ε → 0 uniformly with respect to t ∈ [0, r0), x ∈ Ds.A different approach is required in order to prove a convergence result

for the integral term

I1ε (t, x) =m∑

j=1

∫ t∧τ0

gj(G(p̂ε(s), x, λ(s, x))) αjε(s)ds.

It is quite obvious that the mapping G(p̂ε(t), x, λ) satisfies the equation

(32) G(p̂ε(t), x, λ) = λ +m∑

j=1

∫ t0

χ(s)[0,τ ]gj(x, v̂ε(s, x, λ) α

jε(s)ds


for any fixed x, λ. We now employ a discretization procedure. Let ∆ := 0 =t0 < t1 < · · · < tp = t be a partition of the interval [0, t]. Then

I1ε (t, x) =m∑

j=1

[ p∑k=1

∫ tk∧τtk−1∧τ

gj(x, G(p̂ (s), x, λ(tk−1, x))) αjε(s)ds]

+m∑

j=1

[ p∑k=1

∫ tktk−1

χ(s)[0,τ ]

(gj(x,G(p̂ (s), x, λ(s, x)))

−gj(x,G(p̂ (s), x, λ(tk−1, x))))

αjε(s)ds]

= J1ε (∆, x) + J2ε (∆, x).

The random function λ(t, ω)) is uniformly bounded in t, as a map taking valuesin Xs1 , due to the boundedness of the coefficients of system (20) and the usualestimates for derivatives of holomorphic functions, namely,

(33)∥∥∥∥ ∂λ∂xi (t)

∥∥∥∥s1

≤ R−1 ‖λ‖s01

s0 − s1for any t ∈ [0, r).

Hence

(34) ‖λ(t)‖s1 ≤C ‖λ‖s0 + D

s0 − s1,

with C,D some nonrandom positive constants. Starting with the integral formof (20), we obtain an estimate for the increments of λ,

(35) ‖λ(v)− λ(u)‖s1 ≤ (v − u)C ‖λ‖s0 + D

s0 − s1for any 0 < u < v ≤ r0. On the other hand,

αjε(t) =1ε

∫ t0

e−1ε(t−s)dWj(s),

and

(36) E(αjε(t))2 =

1ε2

∫ t0

e−2ε(t−s)ds =

12ε

(1− e−

2εt)≤ 1

2ε.

By the Cauchy–Holder inequality, the boundedness of ∇λG(p, x, λ) and esti-mates (35) and (36), we get

E|J2ε (∆, x)|2 ≤ C(m)m∑

j=1

p∑k=1

∫ tktk−1

E(αjε(s))2ds E

∫ tktk−1

|λ(s, x)− λ(tk−1, x)|2ds

≤ C(m, s0, s1) t12ε

‖∆‖3,

(37)


where we put ‖∆‖ := max1≤k≤p

(tk − tk−1). Choosing now a partition ∆ with

‖∆‖ ≤ ε, we deduce that J2ε (∆, x) → 0 in L2(Ω) as ε → 0 for t ∈ [0, r0), x ∈Ds1 . By formula (32), the integral term J

1ε (∆, x) may be rewritten as

J1ε (∆, x) =p∑

k=1

[G(p̂ε(tk), x, λ(tk−1, x))−G(p̂ε(tk−1), x, λ(tk−1, x))

]and it is not hard to see, on account of the boundedness of the first partialderivatives of G with respect to p and formula (27), that

J1ε (∆, x) → J1(∆, x) :=p∑

k=1

[G(p̂ (tk), x, λ(tk−1, x))−G(p̂ (tk−1), x, λ(tk−1, x))

],

in L2(Ω) as ε → 0 uniformly with respect to tk ∈ [0, T ], x ∈ D′s0 . For each k,λ(tk−1) may be approximated by step functions of the form

∑i

aikχAik , where

aik are nonrandom elements of Xs1 and Aik ∈ F such that⋃i

Aik = Ω and

Aik, i ≥ 1, are disjoint. For instance, we may consider the sequence

λn(tk−1) =l

2non{

ω∣∣∣ l − 1

2n≤ ‖λ(tk−1, ω)‖s0 ≤

l

2n}

.

We may thus assume that λ(tk−1, x) are nonrandom functions and rewrite

J1(∆, x) =p∑

k=1

∫ tktk−1

χ[0,τ ](s) gj(x,G(p̂ (s), x, λ(tk−1, x))) ◦ dWj(s)

=p∑

k=1

∫ tktk−1

χ[0,τ ](s) gj(x,G(p̂ (s), x, λ(s, x))) ◦ dWj(s)

+p∑

k=1

∫ tktk−1

χ[0,τ ](s)[gj(x,G(p̂ (s), x, λ(tk−1, x)))

−gj(x,G(p̂ (s), x, λ(s, x)))]◦ dWj(s)

=∫ t

0χ[0,τ ](s) gj(x, G(p̂ (s), x, λ(s, x))) ◦ dWj(s) + J3(∆, x).

We now decompose the term J3(∆, x) into the sum of a Riemann integral anda stochastic integral written in Itô’s sense. Using a standard procedure, wededuce that

J3(∆, x) → 0 in L2(Ω) as ‖∆‖ → 0.Hence

I1ε (t, x) →∫ t∧τ

0gj(x, u(s, x)) ◦ dWj(s)


in L2(Ω) as ε → 0 for t ∈ [0, r0), x ∈ Ds1 . Set ûs1 := us0 |[0,r0)×Ds1 . It followsthat ûs1 ∈ C1([0, r0);Xs1). Passing now to the limit in equation (30), we finallyget that ûs1 : [0, r0) × Ω → Xs1 is a solution of system (11) in the sense ofDefinition 1.

4. SPDEs OF PARABOLIC TYPE

The techniques used to derive existence and uniqueness results for nonlin-ear Cauchy-Kowalewska systems allows us to extend the same type of resultsto nonlinear parabolic systems with stochastic perturbations.

The systems we treat have the form

(38)

dui(t, x) = [∆ui(t, x) + fi (t, x, u(t, x),∇u1(t, x), . . . ,∇uN (t, x))]dt

+m∑

j=1

gij(x, u(t, x)) ◦ dWj(t); u = (u1, . . . , uN ),

u(0, x) = u0(x); t ∈ [0, T ], x ∈ Bρ := {x ∈ Rn | ‖x‖ ≤ ρ},where

(i1) {W (t), t ∈ [0, T ]} is a standard m-dimensional Brownian motionon the complete filtered probability space {Ω,F ,Ft,P}, and the filtration{Ft, 0 ≤ t ≤ T} satisfies the usual conditions;

(i2) the mappings fi(t, x, u, p1, . . . , pN ) : [0, T ] × Bρ × RN × RnN → R,1 ≤ i ≤ N , are locally bounded and locally Lipschitz continuous with respectto (u, p), i.e.,

(39) |fi(t, x, u′′, p′′1, . . . , p′′N)−fi(t, x, u′, p′1, . . . , p′N )|≤L(|u′′−u′|+

N∑j=1

|p′′j−p′j |)

for any (t, x) ∈ [0, T ]×Bρ and |u′|, |u′′|, |p′k|, |p′′k| ≤ δ, where L, δ are fixed and,in addition, we assume that there exists a positive constant C such that

|fi(t, x, u, p1, . . . , pN )| ≤ Cfor any (t, x) ∈ [0, T ]×Bρ, |u|, |pk| ≤ δ;

(i3) we have gj(· , u) ∈ C2(Bρ; RN ) and gj(x, ·) ∈ C∞(DN ; RN ) for j =1, . . . ,m, where DN := [−δ, δ]N and, moreover, we assume that the Lie al-gebra Γ(x) generated by the C∞ difussion fields gj(x, ·) is finite dimensional,uniformly with respect to x ∈ Bρ;

(i4) u0 ∈ C2(Bρ;DN ).Definition 2. We say that the pair (u, τ) consisting of a process u(t, x, ω) :

[0, a]×Bρ × Ω → DN and a stopping time τ of the filtration {Ft, 0 ≤ t ≤ T}is a solution of system (38) if

(a) u(· , x) is a continuous and {Ft}-adapted process for any x ∈ Bρ;(b) u(t, ·) ∈ C2(Bρ;DN ), for any t ∈ (0, a];


(c) u fulfills the integral system

u(t, x)= u(σ, x) +∫ t

σ

[∆u(s, x) + f(s, x, u(s, x),∇u1(s, x), . . . ,∇uN (s, x))

]ds

+m∑

j=1

∫ t∧τσ∧τ

gj(x, u(s, x)) ◦ dWj(s),(40)

for any 0 < σ < t ≤ a, x ∈ Bρ.Hypothesis (i3) is used to define a smooth diffeomorphism G(p, x, λ),

as the composition of local flows Gj(t, x, λ) generated by the diffusion vector

fields gj(x, u), 1 ≤ j ≤ m, where p = (t1, . . . , tm) ∈ Im :=m∏

i=1[−ai, ai], x ∈ Bρ,

λ ∈ EN := [−η, η]N .We know that G(· , x, λ) ∈ C∞(Im; RN ), G(p, · , λ) ∈ C2(Bρ; RN ) and

G(p, x, ·) is a C∞ diffeomorphism. Since G is defined on a compact set, wemay assume that all the partial derivatives of order less than three with respectto λi, xk and their inverses are bounded by a positive constant C. We alsosuppose that G

(Im ×Bρ × EN

)⊆ DN .

In addition, using the properties of the associated gradient system, weget a nonsingular matrix A(p) whose components are analytic functions, suchthat we can represent the corresponding gradient system as

∇pG(p, x, λ) =(Y1(x,G(p, x, λ)), . . . , YM (x,G(p, x, λ))

)A(p),

for any p ∈ Im, y ∈ EN , x ∈ Bρ, and {Y1(x, ·), . . . , YM (x, ·)} is a basis in Γ(x).This yields the existence of some analytic vector functions qj(p), 1 ≤ j ≤ m,such that

(41) ∇pG(p, x, λ) qj(p) = gj(x,G(p, x, λ)).

Define the processes p̃ (t) and p̂ (t) as in (17) and (19). Recall that p̂ (t)fulfills the SDE

p̂(t) =m∑

j=1

∫ t∧τ0

qj(p̂ (s)) ◦ dWj(s) =m∑

j=1

∫ t0

χ[0,τ ](s)qj(p̂ (s)) ◦ dWj(s),

where the stopping time τ is defined as in (18). Set S(t, x, λ) := G(p̂ (t), x, λ).It is easily seen, by applying Itô’s stochastic rule of differentiation to theprocess G(p̂ (t), x, λ), that

(42) S(t, x, λ) = λ +m∑

j=1

∫ t0

χ[0,τ ](s) gj(x, S(s, x, λ)) ◦ dWj(s)

for t ∈ [0, T ], x ∈ Bρ, λ ∈ EN . S(· , x, λ) is a continuous and {Ft}-adaptedprocess, S(t, · , λ) ∈ C2(Bρ; RN ) and S(t, x, ·) ∈ C∞(EN ; RN ) .


We are next looking for a solution of system (38) of the form u(t, x) :=S(t, x, α(t, x)), with α(t, x) solving a parabolic differential system with randomparameter. Consider the parabolic system

(43)

∂α

∂t(t, x) = ∆α(t, x) + f̃(t, x, α(t, x),∇α1(t, x), . . . ,∇αN (t, x)),

α(0, x) = u0(x),

where

f̃(t, x, λ, q) :=(

∂G

∂λ

)−1(p̂ (t), x, λ)

[∆G(p̂ (t), x, λ)

+ 2∂2G

∂λk∂xi(p̂ (t), x, λ) qki + f

(t, x, G(p̂ (t), x, λ),

∂G

∂x1(p̂ (t), x, λ)

+∇λG(p̂(t), x, λ) q1, . . . ,∂G

∂xn(p̂ (t), x, λ) +∇λG(p̂ (t), x, λ) qn

)]for (t, x, λ, q) ∈ [0, T ]×Bρ × EN × RnN .

Proposition 2. System (43) admits a unique solution α(t, x) ∈C1,2((0, a] × Bρ; RN ) which is an adapted process with respect to the filtra-tion {Ft}.

Proof. Let f̄ : [0, T ]×Bρ × RN × RnN × Ω → RM be defined as

f̄(t, x, λ, q, ω) := f̃(t, x, λ, q, p̂ (t, ω)).

Obviously,

f̄(t, x, λ, q, ω) = f̃(t, x, λ, q, ω)

for (t, x, λ, q) ∈ [0, T ]×Bρ × EN × RnN , ω ∈ Ω.Consider the nonlinear parabolic system

(44)

∂λ

∂t(t, x) = 4λ(t, x) + f̄(t, x, λ(t, x),∇λ(t, x)),

λ(0, x) = 0; t ∈ (0, T ], x ∈ Bρ.

The components of the mapping f̄(t, x, λ, q) are continuous, locally boundedand locally Lipschitz continuous with respect to the couple (λ, q). A classicalresult from [1] provides a unique solution α(t, x, ω) : (0, a] × Bρ × Ω → RNof system (44). Moreover, the solution α(t, x) can be constructed using an


iterative procedure. Define the sequence

λ0j (t, x) = 0; q0j (t, x) = 0,

λk+1j (t, x) =∫ t

0

[ ∫Rn

f̄j(s, y, λk(s, y), qk(s, y))Γ(t− s, x, y)dy]ds,

qk+1j (t, x) =∫ t

0

[ ∫Rn

f̄j(s, y, λk(s, y), qk(s, y))∇xΓ(t− s, x, y)dy]ds,

where Γ(t, x, y), t > 0, x, y ∈ Rn, x 6= y, stands for the elementary solution ofthe homogeneous heat equation, i.e.,

Γ(t, x, y) = (4πt)−n2 e−

‖y−x‖24t , t > 0, x, y ∈ Rn.

Then (α, q̄) is the limit of the sequence (λ(k), q(k)) as k → ∞ in the Banachspace Cb([0, a] × Bρ; RN × RnN ), and α(t, x, ω) belongs to C([0, a] × Bρ) ∩C1,2((0, a]×Bρ), for a.e. ω.

All the terms of the sequence are adapted processes and the iterativeprocedure used in obtaining the solution allows us to see that the same willhold for the limiting process. �

Suppose, without any loss of generality, that u0(·) ≡ 0. The main resultof this section is given by

Theorem 5. Define the process

u(t, x, ω) = S(t, x, α(t, x, ω)), t ∈ [0, a], x ∈ Bρ.

Then the pair (u, τ) is a solution of system (38), in the sense of Definition 2.In addition, let ũ also be a solution of system (38), defined on [0, a]×Bρ×Ω.Then u and ũ are indistinguishable processes.

Proof. Define Xt(x, ω) as the multidimensional process

Xt(x, ω) =(

p̂ (t, ω)α(t, x, ω)

)∈ RM+n, t ∈ [0, a], x ∈ Bρ.

Applying Itô’s rules to the process G(Xt(x, ω), x) = G(p̂ (t, ω), x, α(t, x, ω)) onthe time interval [σ, t], with 0 < σ < t ≤ a, we get

u(t, x) = G(Xt(x), x) = G(Xσ(x), x) +∫ t

σ∇pG(p̂ (s), x, α(s, x)) dp̂ (s)

+∫ t

σ∇λG(p̂ (s), x, α(s, x))

∂α

∂t(s, x)ds

+12

M∑i,j=1

m∑k=1

∫ tσ

∂2G

∂ti∂tj(p̂ (s), x, α(s)) qik(p̂ (s)) q

jk(p̂ (s))ds


= u(σ, x) +∫ t

σ∇λG(p̂ (s), x, α(s))

∂α

∂t(s, x)ds

+m∑

j=1

∫ t∧τσ∧τ

gj(x, u(s, x)) ◦ dWj(s),

where qk(p) = (qik(p))1≤i≤M , 1 ≤ k ≤ m.The partial derivatives of the first and second order of u with respect to

x are given by∂u

∂xi(t, x) =

∂G

∂xi(p̂ (t), x, α(t, x)) +∇λG(p̂ (t), x, α(t, x))

∂α

∂xi(t, x),

∂2u

∂x2i(t, x) =

∂2G

∂x2i(p̂ (t), x, α(t, x)) + 2

M∑k=1

∂2G

∂λk∂xi(p̂ (t), x, α(t, x))

∂αk∂xi

(t, x)

+∇λG(p̂ (t), x, α(t, x))∂2α

∂x2i(t, x), 1 ≤ i ≤ n.

It follows that

4u(t, x) = 4G(p̂ (t), x, α(t, x)) + 2M∑

k=1

n∑i=1

∂2G

∂λk∂xi(p̂ (t), x, α(t, x))

×∂αk∂xi

(t, x) +∇λG(p̂ (t), x, α(t, x))4 α(t, x), (t, x) ∈ (0, a]×Bρ,

which may be rewritten as

4u(t, x) = ∇λG(p̂ (t), x, α(t, x)) [4α(t, x) + f̄(t, x, α(t, x),∇α(t, x))]

= ∇λG(p̂ (t), x, α(t, x))∂α

∂t(t, x).

We used the fact that α(t, x) is solution of (43). Hence

u(t, x) = u(σ, x) +∫ t

σ4u(s, x)ds +

m∑j=1

∫ t∧τσ∧τ

gj(x, u(s, x)) ◦ dWj(s),

and thus the pair (u, τ) satisfies system (40).G is a continuous mapping and p̂ (t), λ(t, x) are continuous and Ft-

adapted processes for fixed x, so that (a) of Definition 2 is satisfied. Theregularity of u with respect to x follows easily.

The diffeomorphism property of G with respect to λ allows us to obtaina local uniqueness result. The functional equation

ũ(t, x) = G(p̂ (t), x, α), t ∈ [0, a], x ∈ Bρ,admits a unique solution given by

α̃(t, x) := H(p̂ (t), x, ũ(t, x)),


where H stands for the inverse of G with respect to λ, i.e.,

H(p, x, λ) : IM ×Bρ × EN , H(p, x, λ) := GM (−tM , x) ◦ . . . ◦G1(−t1, x)(u),

for p = (t1, . . . , tm) ∈ Im, x ∈ Bρ, u ∈ EN . It follows that α̃ ∈ C([0, a] ×Bρ; RN ) ∩ C1,2((0, a] × Bρ; RN ). We next apply Itô’s formula to the processG(p̂ (t), x, α̃(t, x)). A standard computation yields

∇λG(p̂ (t), x, α̃(t, x))∂α̃

∂t(t, x)=∇λG(p̂ (t), x, α̃(t, x))[4α̃(t, x)+ f̃(t, x, α̃(t, x))]

for t ∈ [σ, a], x ∈ Bρ. We deduce that α̃ solves system (43). We use nowProposition 2 in order to get

α̃(t, x, ω) = α(t, x, ω), a.e. ω ∈ Ω, (∀) t ∈ (0, a], x ∈ Bρ,

and the proof is complete. �

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Received 12 November 2007 Academy of Economic Studies6, Piaţa Romană

Bucharest, [email protected]

and

Romanian Academy“Simion Stoilow” Institute of Mathematics

P. O. Box 1–764Calea Grivitei 21

014700 Bucharest, [email protected]

evolution systems of cauchy-kowalewska and parabolic type with stochastic perturbations · 2010. 9....

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