The Cocycle Property of Stochastic differential Equations Driven by G-Brownian Motion∗

Huijie QIAO

1 Introduction

Let Ω denote the space of all Rd-valued continuous pathswithω0=0,equipped with a uniform convergence topology.If a linear expectationEP,which is induced by the Wiener measureP,is given,the canonical processBt(ω)is ad-dimensional Brownian motion on the probability space(Ω,F,P).HereFstands for a Borelσ- field of Ω.Now,if a sublinear expectation E is given,what is the canonical processBt(ω)on some constructed probability space?It is a G-Brownian motion introduced in[7].The stochastic calculus with respect to the G-Brownian motion has been established(see[7–9]).Relative to the L´evy theorem of the Brownian motion,a martingale characterization of G-Brownian motion has been shown in[10].The BDG inequality for G-stochastic integrals was also established(see[6]).Moreover,the G-Itˆo formula in[8]was obtained and later Gao in[6]extended it by the localization method.

Consider the following stochastic differential equation(SDE,for short)driven by G-Brownian motion:

wherewith the following form:

and

HereATstands for the transposition of matrixA.The second and third integral on the right side of(1.1)will be introduced in Section 2.Ifb,handσsatisfy Lipschitz conditions,Peng[8]showed the existence and uniqueness of the solution to Equation(1.1)in the space(see the definition in Section 2)by the contracting mapping theorem.Under the same conditions,Gao defined the Picard iterative approximation sequence and obtained the unique solution to(1.1)(see[6]).However,many coefficients do not satisfy the Lipschitz condition.Therefore the extension to non-Lipschitz conditions is necessary.Here we do this.The unique solution to(1.1)is constructed through successive approximation.It is worthwhile to mention that non-Lipschitz conditions have been studied in[2].But our assumption is more general than theirs.

Moreover,we study a property of(1.1).Because the property is similar to the cocycle property of SDEs driven by Brownian motion,we also call it the cocycle property.As we know,it is the first time to mention the cocycle property of SDEs driven by G-Brownian motion.

Using the cocycle property,under some special non-Lipschitz conditions,we get a bicontinuous modification of the solution with respect tot,x.

This paper is arranged as follows.In Section 2 we prepare some preliminaries to the readers’convenience.In Section 3,the solution to(1.1)is constructed and its cocycle property is proved.We consider its bi-continuity under some special non-Lipschitz conditions in Section 4.

The following conventions will be used throughout the paper:Cwith or without indices will denote different positive constants(depending on the indices)whose values may change from one place to another.

2 Preliminaries

First of all,we introduce G-expectation(see[5–6]).

Sddenotes the space ofd×dsymmetric matrices.Γ is a given nonempty,bounded and closed subset of Rd×dwhich is the space of alld×dmatrices.lip(Rd)is the set of bounded Lipschitz continuous functions on Rd.|·|denotes the length of a vector in Rn.?·?stands for the Hilbert-Schmidt norm of a matrix.

ForA∈Sd,set

For eachϕ∈lip(Rd),define

whereu(t,x)is the viscosity solution to the following G-heat equation:

and(the existence and uniqueness of(2.1)in the sense of the viscosity solution can be found in[4]).Thenis a sublinear expectation,i.e.,

This sublinear expectation is also called a G-normal distribution on Rdand is denoted byN(0,Σ),where

To well understand the sublinear expectation,we introduce another concept.LetFt:=andbe the collection of all Γ-valued-adapted processes on the interval[0,∞).For each,we denote

Pθdenotes the law of the processunderP.Define

ThenC(·)is a Choquet capacity(see[5]).A setAis polar ifC(A)=0 and a property holds“quasi-surely”(q.s.,for short)if it holds outside a polar set.For eachX∈Fsuch thatEPθXexists for eachSet

and then for all)(introduced in the sequel)

For its proof,refer to[5].

For eacht>0,set

LetHbe a vector lattice of real functions defined on Ω such that Lip(F)⊂Hand if thatX1,···,Xn∈H,theng(X1,···,Xn)∈Hfor eachg∈lip(Rn).

Let E:H?→R be a sublinear expectation onH.Ad-dimensional random vectorXwith each component inHis said to be G-normal distributed under the sublinear expectation E[·]if for eachϕ∈lip(Rd),

is the viscosity solution of the G-heat equation(2.1).

E is called to be a G-expectation if thed-dimensional canonical processis a G-Brownian motion under the sublinear expectation,that is,

(i)B0=0;

(ii)for any

(iii)for anym?1,0=t0

whereIn particular,for anyis a one-dimensional G-Brownian motion.

Next,we only introduce a stochastic integral aboutfor convenience of statement(see[5–6]).

To the G-expectation E,the topological completion of(resp.Lip(F))under the Banach norm E[|·|]is denoted by(resp..E[·]can be extended uniquely to a sublinear expectation onThe extension is also denoted by E.

ForLetp?1 be fixed.Define

whereFo

For eachdenotes the completion ofunder the norm

For eachde fi ne

and then we get a stochastic integral with respect to G-Brownian motion.Besides,the mappingcan be continuously extended toIFor eachthe stochastic integral is defined by

Letbe a sequence of partitions of[0,t]and

The quadratic variation processof the processis defined by

For each fixeds?0,

whereand

De fi ne a mappingas follows:

ThenQ0,Tcan be uniquely extended toWe still denote this mapping by

The following two theorems from[6,Theorems 2.1–2.2]are BDG inequalities for the G-stochastic integral.

Theorem 2.1For andThen there exists acontinuous modificationof X,i.e.,foris continuous and for all t∈[0,T],such that

where0

Theorem 2.2Let p?1and Then there exists a continuous modificationsuch that for any0?s

3 SDEs Driven by G-Brownian Motion

Theorem 3.1Suppose that for p?2,

(1)there exists a function H(t,u):such that

(1a)for fixed t,H(t,u)is continuous nondecreasing with respect to u,

(1b)for T>0,any0

and

(1c)for any constant K>0,the differential equation

has a global solution for any initial value u0;

(2)there exists a function F(t,u):such that

(2a)for fixed t,F(t,u)is continuous nondecreasing in u and F(t,0)=0,

(2b)for T>0,any0

(2c)for any constant K>0,if a non-negative function ϕtsatis fi esfor all t∈R+,then ϕt=0.

Then(1.1)has a unique solution X which is continuous q.s.and for t>0.

Remark 3.1FixT>0 and assume thatb,handσsatisfy,for allx,x1,x2∈Rn,

whereare square integrable andis a continuous,increasing and concave function so that

Under these conditions(H1)and(H2),Bai and Lin in[2,Theorem 3.1]showed the existence and uniqueness of the solution to(1.1).If we chooseandF(t,u)=β2(t)ρ(u),it can be easily justified that(1a)–(1c)and(2a)–(2c)hold forp=2.Therefore,our result is more general than that in[2].

We are now in a position to give the proof of Theorem 3.1.

Proof of Theorem 3.1Letand forn∈N,

∀T>0.First of all,we show that fort

whereutsatisfies

and

Suppose

and

which together with the definition of the G-stochastic integral and(1b)yieldSecondly,by Theorems 2.1–2.2,the H¨older inequality and(1a)–(1b),we get

for allt?T.By the induction method,(3.2)is proved.

Next,we have,by the same deduction as above,

whereLet

It follows from the Fatou lemma and(2a)that

By(2c),we obtain thati.e.,

Then there exists a subsequencesuch that for any

Thus

which implies

Lettingand taking limits on both sides of the above inequality,we get

Now by the H¨older inequality,(2b)and Theorems 2.1–2.2,it holds that

and

Taking limits on both sides of(3.1)inwe attain thatXsatisfies(1.1).

Next,letXandX?be two solutions to(1.1),and then by the same way as above,we obtain that

Next assume thatb,handσare independent oft,ω,and we study the cocycle property of(1.1)under these conditions of Theorem 3.1.The method comes from[1].

For 0?s

By Theorem 3.1 we know that(3.3)has a unique solution and denote it by Φs,t(x,ω).

Lemma 3.1For0?r

ProofBecause Φr,t(x,ω)solves(3.3),it follows from the additive property of G-stochastic integrals(see[7])

However,

By the uniqueness of the solution to(3.3),(3.4)is proved.

Before stating another lemma,we introduce a notation Φ0,t(x,?ω),which solves the following equation:

based onand Theorem 3.1.

Lemma 3.2For

ProofDefineand forn∈N,

Thenare well defined by the proof of Theorem 3.1.First of all,we prove

Assume that(3.6)holds forn−1.Taking a sequence of partitions of[0,t]:one gets by Proposition 5.3.5 in[9]

and by the definitions of G-stochastic integrals,

asThus,by(3.6),Proposition 5.3.5 in[9]and the definitions of G-stochastic integrals,

where these limits hold in

Finally,by the proof of Theorem 3.1 we know thatconverge torespectively inSo,(3.5)is proved by taking the limit to(3.6).

Theorem 3.2(Cocycle Property)For

ProofBy(3.4)–(3.5),we have

4 A Special Case:F(t,u)=Cq(t)ρη(u)

Forwe define a concave function as

ThenCq(t)ρη(u)satisfies(2a)and(2c),whereq(t)is a strictly positive and integrable function on R+.In this section,we consider the special case:F(t,u)=Cq(t)ρη(u).

Lemma 4.1Suppose that b,h and σ satisfy those conditions in Theorem3.1for F(t,u)=Cq(t)ρη(u).Then for any T>0,there are three positive constants C3=C3(p,T),C4=C4(p,T)and C5=C5(p,T)such that for any x,y∈Rnand any s,t∈[0,T],

Its proof is similar to that of Theorem 3.1 and we omit it.

Proposition 4.1Suppose that b,h and σ satisfy those conditions in Theorem3.1for F(t,u)=Cq(t)ρη(u).Moreover,they are independent of t,ω.Then the solution Xt(x)to(1.1)is bi-continuous with respect to t,x.

ProofIn Lemma 4.1 we first choosepsufficiently large and secondlyT=:T0sufficiently small such that

Then by Theorem 31 in[5]

has a bi-continuous modification which is still denoted byXt(x).

Fort∈[T0,2T0],by Theorem 3.2 we have that for alls∈[0,T0],

wheresolves the following equation:

Set

Sinceandare continuous q.s.,we obtain the continuity ofq.s.We still denote the bi-continuous modi fi cation ofbyBy(4.1),we have

Then there is a setwithsuch that for all

In the same way ast∈[T0,2T0],we get thatXt(x)has a bi-continuous modification for

AcknowledgementsThe author is very grateful to Professor Jinqiao Duan and Xicheng Zhang for their valuable discussions,and also wishes to thank the referees for their suggestions and improvements.

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