Stochastic Differential Equations Under Nonlinear Expectation And Related Topics

The nonlinear backward stochastic differential equations(BSDE theory)formulated by Pardoux and Peng[61]has many applications in practice and theory,which range from economics(see e.g.El Karoui,Peng and Quenez[22),PDEs(see e.g.Pardoux and Peng[62].Peng[66])to stochastic control(see Peng[66,671).A solution to a BSDE is a couple of processes(Y,Z)satisfying:Yt=ξ+∫tTf(s,Ys,Zs)ds-∫tTZsdBs,0≤t≤T.Under the Markov assumption,the solution of BSDE can provide probabilistic representations for the solutions of semilinear PDEs(see Peng[66],Pardoux and Peng[63].Fuhrman and Tessitore[28]and Crisan and Delarue[15]).Based on BSDE,Peng[68]introduced the nonlinear g-expectation theory as a nontrivial generalization of classical linear expectations.Indeed,the 9-expectation is described by a class of equivalent probability measures.In sprit of this property,Chen and Epstein[11]studied the stochastic differential recursive utility with drift ambiguity.However,many economic and financial problems involve model uncertainty which is characterized by a family of non-dominated probability measures.Motivated by these questions,Peng[71,72,73]has established systemat.ically a framework of time-consistent sublinear expectation,called G-expectation,by stochastic control and PDE methods.Under the G-expectation framework,a new kind of Brownian motion,called G-Brownian motion,was constructed.The corresponding stochastic calculus of Ito s type was also established.Moreover,by the contracting mapping theorem,Peng[71]and Gao[29]obtained the existence and uniqueness of the solution of stochastic differential equations driven by G-Brownian motion(G-SDEs):(0.0.22)where B＝(B1,...,Bd= is G-Brownian motion and(Bi,Bj)is its cross-variation process,which is not deterministic unlike the classical case.Furthermore,Hu.Ji.Peng and Song[36]obtained the existence and uniqueness of solutions to the following one-dimensional(i.e.,Y is one-dimensional)backward stochastic differential equations driven by G-Brownian motion(G-BSDEs):Yt=ξ+∫tTd(s,Ys,Zs)ds + ∫tTgij(s,Ys,Zs)d<Bi,Bj>s-∫tTZsdBs-(KT-Kt),(0.0.23)where the solution of this equation consists of a triple of processes(Y,Z,K).Note that,compared with the classical case.G-BSDE has an extra non-increasing G-martingale term K.In their accompanying paper[37].the corresponding comparison theorem.Feymann-Ka.c formula were established.For other developments on G-SDE and G-BSDE theory,one can refer to,for instance.Hu,Li,Wang and Zheng[38],Hu,Lin and Hima[43],Li,Peng and Hima[50],Lin[53]and Peng and Song[75].For the further studies on sublinear expectation and G-expectation theory,we refer the reader to Chen,Wu and Li[12],Dolinsky,Nutz and Soner[19],Epstein and Ji[25],Neufeld and Nutz[58],Soner.Touzi and Zhang[83],Xu and Zhang[91]and Zhang[93].This thesis mainly studies several topics on(forward and backward)stochastic differential equations driven by G-Brownian motion.To be precise,we discuss the well-posedness of multi-dimensional backward stochastic differential equations driven by G-Brownian motion,the strong Markov property for the stochastic diferential equations driven by G-Brownian motion,the quasi-continuity property of exit times from an open set of nonlinear semimartingales,which regarding G-SDE as a special case and typical example,and backward stochastic differential equations driven by G-Brownian motion with mean reflection.Let us present the organization of the paper.In Chapter 1,we recall some basic notions and properties of G-expectation,G-Brownian motion,G-stochastic differential equations and G-backward stochastic differential equations.In Chapter 2,we study the well-posedness for a class of multi-dimensional backward stochastic differential equations driven by G-Brownian motion.Multi-dimensional G-BSDE refers to the case that Y in the solution is mlti-dimensional.Since G-expectation is a nonlinear expectation,the linear combination of G-martingales is not a G-martingale anymore.This leads to the difficulty that the method for one-dimensional G-BSDE in[36]can not be applied to solving multi-dimensional G-BSDEs.The existence and uniqueness of solutions are obtained via a contraction argument for Y component and a backward iteration of local solutions.Furthermore.we show that,the solution of multi-dimensional G-BSDE in a Markovian framework provides a probabilistic formula for the viscosity solution of a system of nonlinear parabolic partial differential equations.Chapter 3 is devoted to the research of the strong Markov property for the stochastic differential equations driven by G-Brownian motion.We first extend the deterministic-time conditional G-expectation to optional times.This is done by using the method of a priori estimates and proving a new type of consistency property for conditional G-expectation.The strong Markov property for G-SDEs is then obtained by using the discretization method.Due to the fact that dominated convergence does not hold in general,we apply the Kolmogorov’s criterion for tightness and utilize the properties of conditional G-expectation we constructed.In particular,for any given optional time τ and G-Brownian motion B,the reflection principle for B holds and(Bτ+t-Bτ)t≥0 is still a G-Brownian motion.In Chapter 4,we provide a general theory on the quasi-continuity property for the exit times of nonlinear semimartingales from an open set.The G-SDE situation is a special case and main motivation.Under some additional assumptions on the growth and regularity of the semimartingale,we prove that the corresponding exit time is quasi-continuous if the open set satisfies the exterior ball condition.In this argument,we use an auxiliary function method and use the notion of regular conditional distributions.We also give the characterization of quasi-continuous processes and related properties on stopped processes.In particular,we get the quasi-continuity of exit times for multi-dimensional G-martingales,which nontrivially generalizes the previous one-dimensional result,of Song[86].In Chapter 5.we study the backward stochastic differential equation with mean reflection driven by G-Brownian motion.The main difficulty is that the G-expect.ation is an upper expectation of a family of mutually singular probability measures.Then the strict comparison is not obvious and the dominated convergence theorem does not hold under G-framework.But these two properties are crucial for the construction of the push.We overcome these problems by utilizing the weakly compactness of the probability family and proving the uniformly integrability of G-martingales through capacity theory.The existence and uniqueness of solutions of G-BSDEs with mean reflection are then obtained with the help of a martingale representation type argument and the fixed-point theory.We also consider a more general nonlinear expectation reflection.In the following we present.the main results of this dissertation.1.Multi-dimensional BSDEs driven by G-Brownian motionIn this chapter,we shall consider the following type of n-dimensional G-BSDE on the interval[0,T]:Ytl=ξl+∫tTfl(s,Ys,Zsl)ds+∫tTgijl(s,Ys,Zsl)d<Bi,Bj>s-∫tTZsldBs-(KTl-Ktl),1 ≤ l ≤ n,where fl(t,ω,y,z),gijl(t,ω,y,z):[0,T]× ΩT ×Rn × Rd→R,(?)1 ≤ l ≤ n,satisfy:(H1)there is some constant β>1 such that for each y,z,fl(·,·,y,zl),gijl(·,·,zl)∈MGβ(0.T).(H2)there exists some L>0 such that,for each y1,y2∈Rn,z1l,z2l∈ Rd,|fl(t,y1,z2l|+(?)|gijl(t,y1,z1l)-gijl(t,y2,z2l)|≤L|y1-y2|+|z1l-z2l|).We first research the existence and uniqueness theorem of the local solution to G-BSDE(0.0.24).Indeed,we haveTheorem 0.1.Assum.e that(H1)—(H2)hold for some β>1.Then there exists a constant 0<δ≤T depending only on T,G,n,β and L such that for any h ∈(0,δ],t ∈[0,T-h]and given ζ∈LGβ(Ωt+h;Rn),the G-BSDE on the interval[t,t+ h]Ysl =ζl+ ∫st+hfl(r,Yr,Zrl)dr + ∫st+hgijl(r,Yr,Zrl)d<Bi,Bj>r-∫st+hZrldBr-(Kt+hl-Ksl),1≤l≤n,(0.0.25)admits a unique solution(Y,Z,K)∈ SGα(t,t+ h;Rn)× HGα(t,t+ h;Rn×d)× AGα(t,t +h;Rn)for each 1<α<β.Moreover,Y ∈ MGβ(t,t+h;Rn).In order to prove Theorem 0.1,we consider the following G-BSDE on the interval[t,t +h]:YsU,l =ζl+ ∫st+hfl,U(r,YrU,l,ZrU,l)dr+ ∫st+hgijl,U(r,YrU,lZrU,l)d<Bi,Bj>r-∫st+hZrU,ldBr-(Kt+hU,l-KsU,l),1≤l≤n,where U∈MGβ(t,t+h;Rn),ζ∈(Ωt+h;Rn),h∈[0,T-t]and for ψ=f,gij,ψl,U(t,yl,zl):=ψl(t,Ut1,…Utl-1,yl,Utl+1,…Utn,zl):[0,T]×ΩT×R×Rd→R.Denote XU =(XU,1,…,XU,n)for X = Y.Z,K.ThenLemma 0.1.Suppose assumptions(H1)-(H2)hold for some β>1.Then,for any 1<α<β the G-BSDE(0.0.26)has a uniqrue SGα(t+ t+h;Rn)× HGα(t,t + h;Rn×d)×R AGα(t,t+ h;Rn)-solution(YU,ZU,KU).Moreover,YU∈Gβ(t,t+h;Rn)In the view of Lemma 0.1,we can define the solution map Γ:U → Γ(U)from MGβ(t,t + h;Rn)to MGβ(t,t + h;Rn)byΓ(U):=YU,(?)U ∈MGβ(t,t+h;R).After showing that the solution map is a contraction,we prove by the Picard iteration the existence and uniqueness Theorem 0.1 f local solutions.The well-posedness for G-BSDE(0.0.24)on the whole interval[0,T]is obtained by a backward iteration.Theorem 0.2.Suppose(H1)—(H2)are satisfied for some β>1.Then for any 1<α<β,the G-BSDE(0.0.24)has a unique solution(Y,Z,K).)∈SGα(0,T;Rn)×HGα,(0,T;Rn×d)×AGα(0,T;Rn).Moreover.Y ∈ MGβ(0,T;Rn).We also obtain the comparison theorem for multi-dimensional G-BSDEs(0.0.24).Consider the following two G-BSDEs on the interval[0,T]:Ytl=ξl+∫tTfl(s,Ys,Zsl)ds+∫tTgijl(s,Ys,Zsl)d<Bi,Bj>s-∫tTZsldBs-(KTl-Ktl),1≤l≤n,Ytl=ξ+∫tTfl(s,Ys,Zsl)ds+∫tTgijl(s,Ys,Zsl)d<Bi,Bj>s-∫ZsldBs-(KTl-Ktl),1≤l≤<n.Then we haveTheorem 0.3.Suppose that fl(t,y,zl),fl(t,y,zl),gijl(t,y,zl),gijl(t,y,zl)satisfy(H1)-(H2)and ξ,ξ∈LGβ(ΩT)for some β>1.Assume the following conditions old:(ⅰ)for any 1≤l≤n,and for each.t ∈[0.T],zl∈ Rd and y,y ∈ Rn satisfying yj ≥yj for j≠l and yl = yl,it holds that fl(t,y,zl)≥fl(t,y,zl),[gijl(t,u,zl]ij=1d≥[gijl(t,y,zl)]i,j=1d’(ⅱ)ξ≥ξ.Then we have Yt ≥ Yt for each t∈[0.T]Finally we study the connection between t.he multi-dimensional G-BSDE and the system of fully nonlinear PDEsFs.First,for any given t∈[0,T]and η∈ E LGp(Ωt;Rk),p ≥ 2,we introduce the fllowing G-SDEs:where b(s,x),hij(s,x):[0,T]×Rk→Rk and σ(s,x):[0,T]× Rk→Rk×d are deterministic cont.inuous functions satisfying:(H3)hij =hji for 1≤i,j≤ d,and there exists a positive constant L such that|b(t,x1)-b(t,x2)|+(?)hij(t,x1)hij(t,x2)|+(?)|σ(t,x1)-σ(t,x2)|≤L|x1-x2|.Next we consider the following n-dimensional G-BSDEs on the interval[t,T]:Yst,η;l=φl(XTt,η)+∫sTfl(r,Xrt,η,Yrt,η,Zrt,η;l)dr+ ∫sTgijl(r,Xrt,η,Yrt,η,Zrt,η;l)d<Bi,Bj>r-∫sTZrt,η;ldBr-(KTt,η;l-Kst,η;l),1≤l≤n,(0.0.28)where the deterministic continuous functions φl:Rk→R,fl,gijl=gjil:[0,T]×Bk × Rn × Rd→R,1 ≤ l ≤ n,satisfy the following assumptions:(H4)There exists a constant L ≥ 0 such that|φl(x1)-φl(x2)|+|fl(t,x1,y1,z1l)-f(t,x2,y2,z2l)|+(?)|gijl(t,x1,y1,z1l)-gijl(t,x2,y2,z2l)|≤L(x1-x2|+|y1-y2|+|z1l-z2l).For each(t,x)∈[0,T]× Rk,we define the following deterministic function u(t.x):=Ytt,x,for each(t,x)∈[0,T]×Rk,(0.0.29)and have:Theorem 0.4.The function u is the viscosity solution of the following system of parabolic PDEs:where Fl(A,p,r,x,t):=G(σT(t,x)Aσ(t,x)+2[<p,hij(t,x)>]i,jd=1+2[gijl(t,x,r,σT(t,x)p)]i,j=1d)+<b(t,x),p>+fl(t,x,r,σT(t,x)p),for(A,p,r,x,t)∈S(k)× Rk ×Rn×Rk ×[0,T].3.Strong Markov property for SDEs driven by G-Brownian motionIn this chapter,we study the strong Markov property for G-SDEs.We first provide a construction of the conditional G-expectation Eτ+ for any optional time τ and study its properties.The mapping τ:Ω →[0,∞)is called a stopping time if {τ≤<t}∈ Ft for each t>0 and an optional time if {τ<t}∈Ft for each t ≥ 0.For each optional time τ.we define the σ-field Fτ+:+{ A∈F:A ∩:{τ<t} ∈Ft,(?)tτ≥ 0} = {A∈F:A∩{τ≤t}∈Ft+,(?)t ≥ 0},where Ft =∩s>tFs.Let τ be an optional time.For each p ≥ 1,we set LG0.p,τ+(Ω)=X=(?)ξiIAi:n ∈ N.{Ai}in=1 is an Fτ+-partition of Ω,ξi∈LGp(Ω),i=1,…,n}and denote by Lp,τ+(Ω)the completion of LG0.p,τ+(Ω)under the norm ‖·‖p.We define the conditional G-expectation Eτ+:LG0.p,τ+(Ω)→LG0.p,τ+(Ω)∩L0(Fτ+),and have:Proposition 0.1.The conditional expectation Eτ+:LG1,τ+(Ω)∩L0(Fτ+)satisfies the following properties:for X,Y ∈ LG1,τ+(Ω)，(ⅰ)Eτ+X≤Eτ+[Y],forX≤Y;(ⅱ)Eτ+[X+Y]≤Eτ+[Y]+Eτ+[Y];(ⅲ)E[Eτ+[X]]=E[X].We also give some further properties of Eτ+ on LG0.p,τ+(Ω).Proposition 0.2.The conditional expectation Eτ+:LG0.p,τ+(Ω)→LG1,τ+(Ω)∩L0(Tτ+)satisfies the followi’ng properties:(ⅰ)If Xi ∈LG1,τ+(Ω),i = 1,…,n and {Ai}i=1n is an Fτ+-partition of Ω,then Eτ+(?)XiIAi]=(?)Eτ+[Xi]IAi;(ⅱ)If τ and σ are two optional times and X ∈ LG1,τ+(Ω),then Eτ+[X]I{τ≤σ}=E(τ^σ)+[XI{τ≤σ];(ⅲ)IfX ∈ LG1,τ+＋(Ω),then E(τ^T)+[XI{τ≤T}]→Eτ[X]in L1，as T → ∞；(ⅳ)If{τn}n=1∞,τ are optional times such that Tn →τ uniformly,as n → ∞ and X ∈ LG1,τ0+(Ω),where τ0:= τ ^(^n=1∞τn),the Eτn+[X]→Eτ+[X]in L1,as n →∞;in particular,if τn↓τ uniformly,as n ∞ and X ∈ LG1，τ+(Ω),then Eτn+[X]→ Eτ+[X]in L1,as n→ ∞.Proposition 0.3.The conditional expectation Eτ+ satisfies:(ⅰ)If X ∈LG1，τ+(Ω),and η,Y∈LG1，τ+(Ω),∩0(Fτ+)such that η is bounded,then Eτ+[ηX+ Y]=η+Eτ+[X]+ η-Eτ+[-X]+ Y;(ⅱ)If η∈LG1，τ+(Ω;Rd)∩L0(Fτ+;Rd),X ∈ LG1，τ+(Ω;Rn)and φ∈Cb.Lip(Rd+n),then Eτ+[φ(η,X)]=Eτ＋[φ(p,X)]p=η.As an application,we give the following reflection principle for G-Brownian motion.Theorem 0.5.Let τ be an optional time.Then Bt:2Bt^τ-Bt=Bt^τ-(Bt-Bτ）I{t>τ},for t≥0,is still a G-Brownian motion.Based on Eτ+,we then establish the following strong Markov property for G-SDEs.Theorem 0.6.Let(Xtx)t≥0 be the solution of G-SDE and τ be an optional time.Then for each φ ∈ Cb.Lip(Rm×n)and 0 ≤ t1 ≤ …≤ tm =T’<∞,we have Eτ+[φ(Xτε1τ…,Xτ+tmx)]=E[φ(Xt1y,…,Xtmt]y=Xτx.(0.0.31)Taking n = d,x = 0,b= hij = 0,σ:=(σ1,…,σd)= Id×d in the above theorem,we immediately have the following strong Markov property for G-Brownian motion which says that G-Brownian motion starts afresh at an optional time.Corollary 0.1.For each φ∈ Cb.Lip(Rm×d),0≤t1≤ …≤tm<+∞,m∈N,we ave Eτ+[φ(Bτ+t1-Bτ...,Bτ+tm-Bτ)]= E[φ(Bτ+t1-Bτ,…,Bτ+tm-Bτ)]=E[φ(Bt1,…Btm)].We finally give an application.Let(Bt)t≥o be a 1-dimensional G-Brownian otion such that σ2:=-E[-B12]>0(non-degeneracy).Let a ∈R be given.For ach ω ∈ Ω.define the level set Lω(a):={t≥0:Bt(ω)=a}.((0.0.32)sing the strong arkov property for G-Brownian motion,we can obtain the fol-owing theorem.Theorem 0.7.For q.s.ω∈Ω the level set Lω(a)has no isolated point in[0,∞)..3.Exit time problems and their applicationsWe shall study the quasi-continuity property of exit times for nonlinear semimartingales on a general sublinear expectation space.Let Bt(ω):=ωt for ω∈Ω t≥ 0 be the canonical process and Ft:=σ｛Bs s≤ t} for t≥ 0 be the natural filtration of B.We denote F:(Ft)t≥ 0.Let P be a family of probability measures on(Ω,B(Ω)).We set L(Ω):= {X∈B(Ω):EP[X]exists for each P∈P}.We define the corresponding upper expectation by E[X]:= supp Ep[[X],for X∈L(Ω).(0.0.33)For this P,we define the corresponding upper capacity c(A)supp P(A),A ∈B(Ω).Definition 0.1 An F-adapted prrocess Y =(Yt)t≥ 0 is called a P-martingale(P_spermartingale,P-submartingalele P-emimartingale resp.)if it is a martinale supermartingale,submartingale,semimartingale resp.)under each P ∈P.Let Y be a d-dimensional continuous P-semimartingale under a given weakly compact family P of probability measures.Assume that,under each P ∈ P,we have the decomposition Yt = MtP + AtP,where MtP is a d-dimensional continuous local martingale and AtP is a d-dimensional finite-variation process.We also denote by<Y>P＝<MP>P the quadratic variation under P.For each set D(?)Rd.we define the exit times of Y from D by TD(ω):=inf{t≥ 0:Yt(ω)∈ Dc},for ω ∈ Q.Definition 0.2.An open set O is said to satisfy the exterior ball condition at x ∈(?)O if there exists an open ball U(z,r)with center z and radius r such that U(z,r)(?)Oc and x ∈(?)U(z,r).An open set O is said to satisfy the exterior ball condition if every boundary point x ∈(?)O satisfies the exterior ball condition.Given an open set Q in Rd,we denoteΩω={ω’∈Ω:ωt’=ωt on[0.τQ(ω)]},for each ω∈Ω.(0.0.34)First,we shall mainly deal vwith nonlinear semimartinales Y possessing a local growth condition at the boundary:(H)For each P ∈ P，there exists a P-null set N such that,if ω∈ Nc satisfiesτQ(ω)<∞,then there exist some stopping time σω and constants λω,εω>0 so that(ⅰ)σω(ω’)>0 for ω’∈Ωω:(ⅱ)For ω’∈ Nc ∩Ωω,on the interval[0,σω(ω’)^(τQ(ω’)-τQ(ω’))],it holds that d<MP)τQ(ω)+t(ω’)≥λωtr[d<MP>τQ(ω)+T(ω’)]Id×d,tr[d<AMp>τQ(ω)+t,(ω’)]≥εω|dAPτQ(ω)+t(ω’)|,and tr[d<MP)τQ(ω)+t(ω’)]>0.]Moreover,these three quantities σω,λω and εω can depend on P,ω and aresupposed to be uniform for ω’ ∈Nc∩Ωω.The following theorem establish the quasi-continuity for exit times of nonlinear semimartingales.Theorem 0.8.Let Q be an open set satisfying the exterior ball condition and letΩωbe defined as in(0.0.34).Suppose that Y is quasi-continuous and satisfies condition(H).Then for any δ>0,there exists an open set O O(?)such that c(O)≤ δ and on Oc,we have:(ⅰ)τQ is lower semi-continuous and τQ is upper semi-continuous:(ⅱ)τQ =τQ.In general,we can get the quasi-continuity by a truncation manipulation as follows.Corollary 0.2.Let Y，Q be as in the above theorem.(ⅰ)If X is a quasi-continuous random variable.,then τQ ^ and τQ ^ X are both.quasi-continuous.(ⅱ)If X∈LC1(Ω),then τQ ^ X and τQ ^ X both belong to LC1(Ω).Then we give a characterization theorem on the quasi-continuity of processes as well as some related properties of stopped processes.Theorem 0.9.Let X:Ω x[0,∞)→ be process.(ⅰ)X has a quasi-continuous version on Ω ×[0,T]if and only if we can find a sequence Xn ∈ C(Ω x[0,T)such that,for each.ε>0.c({ sup |Xtn-Xt|>ε})→ 0,as n→∞.(0.0.35)Moreover,we can choose this version to be continuous in t ∈[0,T],for eachω)∈ Q.(ⅱ)X has a quasi-continuous version on Ω ×[0,∞)if and only if for each T>0,there exists a sequence Xn ∈ C(Ω ×[0,T)Ssuch that(0.0.35)holds.Also,this version can be choosen to be continuous in t∈[0.∞),for each ω∈Ω.The following two results concerns the quasi-continuity of stopped processes.Proposition 0.4.Let X=(Xt)t∈[0,∞)be a process.The random variable Xτ is uasi-continuous if one of the following condition holds:(ⅰ)X is a quasi-continuous process on Ω ×[0,T]and τ ≤T is a quasi-continuous stopping time.(ⅱ)X is a quasi-continuous process on Ω ×[0,∞)and τ:Ω → R+ is a quasi-continuous stopping time.Proposition 0.5.Let X =(Xt)t∈[0,∞)be a process.We have(ⅰ)The process(Xτ^t)t∈[0,T]is quasi-continuous on Ω ×[0,T]if X is and τ is a quasi-continuous stopping time.(ⅱ)The process(Xτ^t)t∈[0,∞)is quasi-continuous on Ω ×[0,∞]if X is and τ is a quasi-continuous stopping time.The above characterization theorem contains the following three typical pro-cesses in the G-expectation space.Proposition 0.6.We have:(ⅰ)G-martingale M has a quasi-continuous version on Ω ×[0,∞).(ⅱ)If η∈ MG1(0,T)(∩T>0MG1(0,T)resp.),then the process At:f∫0tηsds has a quasi-continuous version on Ω×[0,T](Ω×[0,∞)resp.).(ⅲ)If η∈ MG1(0,T(∩T>0MG1(0,T)resp.),then the process At:= f∫0tηsds<Bi,Bj>s has a quasi-continuous version on Ω×[0,T](Ω×[0,∞)resp.).By the above proposition and optional sampling theorem for G-martingale,we derived that G-martingale stopped at a quasi-continuous stopping time is still a G-martingale.Corollary 0.3.Let τ be a quasi-continuous stopping time.If(Mt)t≥0 is a G-martingale(symmetric G-martingale resp.),then(Mt^τ)t≥0 is still a G-martingale(symmetric G-martingale resp.).We also have a regularity theorem for the stopping of stochastic integralsProposition 0.7.Let τ≤ T be a quasi-continuous stopping time.Then for each p ≥ 1,we have I[0,τ]∈MGp(0,T].(0.0.36)4.BSDEs with mean reflection driven by G-Brownian motionWe shall consider the following type of G-BSDE with mean reflection,i.e.,the G-expectation of the loss function of component Y in the solution is forced to satisfy a running constraint:Our aim is to find a quadruple of processes(Y.Z,K.R),satisfying equation(0.0.37).The parameters of the G-BSDE with mean reflection are the terminal condition ξ.the generators(or drivers)f,gij as well as the running loss function l.Note that the BSDE with mean reflection may have infinite many flat solutions with random R.Thus we shall study the deterministic solution of G-BSDEs with mean reflection.We denote by AD the closed subset of SG1(0.T)consisting of non-decreasing deterministic processes R with R0 = 0.Definition 0.3.A quadruple of processes(Y,Z,K.R)∈GGα×AD for some α>1 is said to be a deterministic solution to the G-BSDE(0.0.37)with mean reflection if equation(0.0.37)holds true.A solution is said to be "flat" if moreozver R increases only when needed,i.e.∫0TE[l(t,Yt)]dRt= 0.(0.0.38)In the sequel,we are going to study the existence and uniqueness theorem of flat solutions of equation(0.0.37)under the following standard running assumptions:(Hξ)There is some constant β>1 such that ξ is in LGβ(ΩT)and E[l(T,ξ)]≥ 0.(Hl)The running loss function l:ΩT ×[0,T × R→R satisfies the following prop-erties:1.(t,y)→l(t,y)is uniformly continuous,uniformly in ω,2.(?)t ∈[0.T],y→ l(t,y)is strictly increasing.3.(?)t ∈[0.T],(?)y ∈ R,l(t,y)is in LG1(ΩT)and E[lim l(t,y)]>0,4.(?)t ∈[0.T],(?)y ∈ R,l(t,y)≤C(1+ |y|)for some constant C ≥ 0.Due to the complicated structure of G-expectation,in this paper we shall consider two cases of drivers:(Ⅰ)f is deterministic linear dependence on y and gij does not depend on y,(Ⅱ)f and gij do not depend on z.For simplicity we will always assume gij = 0,and the similar results still remain true for the gene.ral case.The driver f in Case I has the following structure:(?)(0.0.39)where at is a deterministic and bounded Borel measurable function.The generator f is supposed to satisfy the following assumption:(Hf’)The driver f:[0,T]× ΩT × Rd→R satisfies the following properties:1.for some constant β>1,(f·,z)is in MGβ(0,T)for each z,2.there exists some constant A>0 such that,for all t E[0,T]and z1,z2∈ Rd,|f(t,z1)-f(t,z2)| ≤λ|z1-z2|.In order to construct a solution to G-BSDE with mean reflection,we need to define the operator Lt:LG1(ΩT)[0,∞),t ∈[0.T],by Lt:X→inf{x≥0:E[l(t,x+X)≥0}.ThenProposition 0.8.We have:(ⅰ)for each(t,x)E[0,T]× R,l(t,x + X)∈ LG1(ΩT)(ⅱ)the map x →l(t,x +X)is continuous under the norm ‖·‖ LG1,in particular x→E[l(t,x + X)]is continuous,moreover x →E[l(t,x + X)]is strictly increasing,(ⅲ)the map t →(t,Et[X]+ ∫0tηudu)is continuous under the norm ‖·‖ LG1,in particular t→ E[l[t,Et[x]+ ∫0tηudu)]is continuous.Based on the above proposition,we prove the following existence and uniqueness theorem.Theorem 0.10.Suppose that(Hξ)-(Hf’)-(Hl)hold.Then,for each 1<α<β.the G-BSDE(0.0.39)with mean.reflection has a unique deterministic flat solution(Y,Z,K,R)∈GGα×AD.We also have the minimality of the deterministic flat solution.Proposition 0.9.Suppose th.at(Hξ)-(Hf’)-(Hl)hold.Then a deterministic flat solution(Y,Z,K)is minimal,i.e..Y-component of the solution is minimal among all the deterministic solutions of the G-BSDE(0.0.39)with mean reflection.The driver f in Case Ⅱ does not depend on z,i.e.,(?)where the generator f is supposed to satisfy the following assumption:(Hf")The driver f:[0,T]× ΩT × →R has the following properties:1.for some constant β>1,f(·,y)is in MGβ(0.T)for each y,2.there exists some constant A>0 such that,for all t ∈[0,T],|f(t,y1)-f(t,y2)|≤λ|y1-y2|,(?)y1,y2∈ R.Based on a contraction argument,we haveTheorem 0.11.Assume that(Hξ)-(Hf’)-(Hl)hold.Then for each 1<α<β,the G-BSDE(0.0.40)with mean reflection has a unique deterministic flat solution(Y.Z,K.R)∈6∈GGα×AD.[0[0,T].We also extend the results of the G-BSDE with mean reflection to the case of nonlinear expectation reflection,i.e.,Here E is a nonlinear expectation dominated by G-expectation E,i.e.,E[X]-E[Y]≤ E[X-Y],(?)X,Y ∈ LG1(ΩT).(0.0.42)By a modification of previous arguments,we also have the well-posedness for G-BSDE with mean nonlinear expectation reflection.Theorem 0.12.Assume that(Hξ)-(Hf’)-(Hu)hold in Case I or(Hξ)-(Hg")-(Hl)-(Hl’)hold in Case Ⅱ.Furthermore,suppose that E[l(T,ξ)]≥ 0 and lim E[l(t,x)]>0.Then in both cases,for each 1<α<β,the G-BSDE(0.0.41)with nonlinear expectation reflection has a unique deterministic flat solu-tion(Y,Z,K,R)∈GGα× AD.