Reflected Backward Stochastic Differential Equations Under Nonlinear Expectations And Their Applications

In 1990,Pardoux and Peng[56]first considered the following type of backward stochastic differential equations(BSDE for short):Yt = ? +?tT(s,Ys,Zz)ds-?tTZsdWs,where g is a given progressively measurable and Lipschitz continuous function,called the generator;? is a given FT measurable and square integrable random variable,called the terminal value.They proved that under these conditions,there exists a unique pair of adapted processes(Y,Z)satisfying the above BSDE.This fundamen-tal work is a powerful tool in solving the following problem,such as mathematical finance([23]),stochastic control problem([59])and partial differential equations(PDE for short,see[57]).Using the solution of BSDE,Peng[60]extended the classical linear expectation to the nonlinear case and defined a new expectation,called g-expectation.This expectation satisfies almost all properties of classical expectation except linearity.Peng[62]introduced a g-supermartingale associated with g-expectation,which is the generalization of the classical supermartingale.Similar with the classical case,Peng obtained the decomposition theorem of Doob-Meyer's type of this nonlinear super-martingale.It plays a key role in solving the representation theorem for filtration consistent nonlinear expectations(see[12]).In 1997,El Karoui et al.[22]studied the reflected backward stochastic differ-ential equations(RBSDE for short).The reflection keeps the solution above a given continuous process,called the obstacle.For this purpose,an additional increasing process should be added in this equation to push the solution upwards.Besides,this process is chosen in a minimal way such that it satisfies the Skorohod condi-tion.The reflected BSDE can.be applied to optimal stopping,American option pricing,Dynkin games and the obstacle problem for PDEs(see[15],[22],[29],[30]).It is important to note that there exists some limitations in the applications of classical BSDE and RBSDE theory.For example,the involved uncertain probability measures have to be absolutely continuous when they are applied to the financial problems under Kightian uncertainty.Hence,the classical results cannot be used to solve the problems in a uncertain volatility model.Besides,the classical BSDE can only provide a probabilistic interpretation of a PDE for quasilinear but not for fully nonlinear case.Motivated by these facts,Peng systematically established a time consistent nonlinear expectation theory,called the G-expectation theory,which can be applied to solve the problems of model uncertainty and fully nonlinear PDEs.This thesis is based on the G-expectation framework and consists of four main topics.In the first one,we extend the definition of supermartingale to the G-expectation framework and we get the decomposition theorem for this new kind of supermartingales.The second one focuses on 1-dimensional reflected backward stochastic differential equations driven by G-Brownian motion(reflected G-BSDEs for short)with a lower obstacle.We obtain the uniqueness and existence of the solutions,the comparison theorem and some applications.The third one is to study 1-dimensional reflected G-BSDE with an upper obstacle.Different from the classical case,this problem is significantly different from the lower obstacle case under G-framework since there is a decreasing G-martingale in the G-BSDE.In the last one,we consider a stochastic optimal control problem where the cost function is defined by the solution of a reflected G-BSDE with a lower obstacle in a Markovian frame-work.Then we establish the dynaamic programming principle(DPP for short)for the value function and prove that the value function is the unique viscosity solution of the related Hamiltion-Jacobi-Bellman-Isaac(HJBI)equation.In this framework,all backward stochastic differential equations are considered under a family of prob-ability measures.The most important and difficult point of G-expectation theory is that this family of probability measures are non-dominated and they could be mutually singular.In these regards,the G-expectation theory shares many similari-ties with the quasi-sure stochastic analysis of Denis and Martini[18]and the second order BSDE theory of Soner,Touzi and Zhand[78].Let us explain the organization of the manuscript more precisely.In Chapter 1,we introduce some basic notions and results of G-expectation and backward stochastic differential equations driven by G-Brownian motion briefly.All the results of the paper is established under G-framework.In Chapter 2,first we introduce a nonlinear operator induced by the solution of G-BSDE and extend the notion of nonlinear supermartingales.Then we prove that the decomposition theorem for this more general supermartingales.In par-ticular,we could get the Doob-Meyer decomposition for G-supermartingales.We need to apply an approximation method via penalization to prove the existence of decomposition.It is worth pointing out that the dominated convergence theorem does not hold under G-expectation.Besides,a bounded subset in MGp(0,T)is not necessarily weakly compact.The main difficulty lies in this problem is to prove the convergence property of the penalized sequence in appropriate sense.For this purpose,we need to use the property on uniform continuity of SGp(0,T)(For the definition of MGp(0,T)and SGp(0,T),we may refer to the first chapter).At last of this chapter,we present the relationship between the nonlinear supermartingales and the corresponding parabolic PDEs.In Chapter 3,we study reflected G-BSDE with a lower obstacle.In contrast with the classical case,we should change the Skorohod condition to the so called martingale condition so as to ensure the uniqueness of the solution.To prove the existence,we construct a family of penalized G-BSDEs.In the third and the fourth section,we present the a priori estimates and convergence properties of reflected G-BSDE and penalized G-BSDEs respectively.Then we could get the uniqueness and existence of solutions and the comparison theorem.In the fifth section,we consider the reflected G-BSDE under the Markovian framework and establish the relation between the solution and a typical kind of fully nonlinear parabolic PDEs with constraints.Finally in the last section,these theoretic results are applied to solve the pricing problem for American options under volatility uncertainty.In fact,Matoussi,Possamai and Zhou[52]tried to solve the second order reflected BSDE but they found a mistake in the minimal condition and in[53]they proposed a new minimal condition for correction.This condition is far more complicated than the martingale condition under G-framework.In Chapter 4,reflected G-BSDE with an upper obstacle is discussed.Due to the appearance of the decreasing G-martingale,the reflected G-BSDE with an upper obstacle is significantly different from the lower obstacle case.We could only prove the existence using approximation method via penalization.But we cannot get the a priori estimates without which makes it difficult for us to prove the uniqueness of the solution.However,by a variant comparison theorem,we show that the solution constructed by the above procedure is the largest one.We may consider that the solution is unique in this sense.In Chapter 5,we study the stochastic recursive optimal control problem with obstacle constraints under G-framework.Here,the value function is defined by the essential infimum of the solutions to a class of reflected G-BSDEs with a lower obstacle.Different from the classical case,the essential infimum of a family of random variables in the "quasi-surely"sense may not exist.Thus in the third section,we prove that the value function is well-defined.Moreover,we claim that it is a continuous deterministic function and satisfies the dynamic programming principle.In the last section,we show that the value function is the unique viscosity solution of the obstacle problem for the corresponding HJBI equations.Here is an overview of the main results of this dissertation.1.Decomposition Theorem for Supermartingales under G-expectationIn this chapter,main results are established on the space SG?(0,T)with ?>2.The definition of this space can be found in Hu et.al[33,34].First,we introduce the definition of general nonlinear Ff,g-supermartingales.For simplicity,consider the following BSDE driven by 1-dimensional G-Brownian motion(the results still hold in the multi-dimensional case):YtT,???+?tTf(s,YsT,?,ZsT,?)ds + ?tTg(s,YsT,?,ZsT,?)d<B>s-?tTZsT,?dBs-(KTT,?-KtT,?),(1)where the generators f and g satisfy the following conditions:(H1)There exists some ?>2 such that for any y,z,f(·,`,y,z),g(·,·?y,z)MG?(0,T);(H2)There exists some L>0 such that|f(t,y,z)-f(t,y',z')| + |g(t,y,z)-g(t,y',z')| ? L(|y-y'|+|z+z'|).For each ? ? LG?{?T)with>2,we define Et,Tf,g[?]:=YtT,?Then we can give the definition of Ef,g-supermartingales.Definition 1.A process {Yt}t?[0,T]is called an Ef,g-supermartingale,if for each t?T,Yt?LG?(?t)with ?>2 and Es,tf,g[Yt]? Ys,(?)0 ? s ?t ?T.In particular,if g?0,we denote this operator by Et,Tf[·].Similarly,we can define the corresponding Ef-supermartingales.For convenience,we mainly focus on the Ef-supermartingales.Proceeding sim-ilarly the procedures in Peng[62],we construct the following family of penalized G-BSDEs,where {Yt}t?[0,T]? SG?(0,T)is a Ef-supermartingale,ytn= YT+?tTf(s,ysn,zsn)+ n?tT(Ys-yxn)ds-?tTzsndBs-(KTn-Ktn).(2)By the representation of the nonlinear conditional expectation Et,Tf[·]and the results in classical case(see[62]),we have for all n = 1,2,…,Yt>ytn Moreover,by the continuity property of the elements in SG?(0,T),we obtain the following convergence property:(?)In summary,the following decomposition theorem for Ef-supermartingales of Doob-Meyer's type is obtained.Theorem 1.Let Y =(Yt)t?[0,T]?SG?(0,T)be an Ef-supermartingale with ?>2.Suppose that f satisfies(HI)and(H2).Then(Yt)has the following decomposition Yt = Y0-?0tf(s,Ys,Zs)ds+?0t ZsdBs-At,q.s.,where {Zt} ? MG2(0,T)a,nd {At} is a continuous nondecreasing process with A0=0 and A ?SG2(0,T).Furthermore,the above decomposition is unique.By applying the same method,we derive the following result for the case where g?0:Theorem 2.Let Y =(Yt)t?[0,T]?SG?(0,T)be an Ef,9-supermartingale with ?>2.Suppose that f and g satisfy(H1)and(H2).Then(Yt)has the following decompo-sition Yt=Y0-?otf(s,Ys,Zs)ds-?0tg[s,Ys,Zs)d<B>s + ?0tZsdBs-At,q.s.,where {Zt} ? MG2(0,T)and {At} is a continuous nondecreasing process with AO = 0 and A ? SG2(0,T).Furthermore,the above decomposition is unique.Then we present the relationship between the Ef-supermartingales and the fully nonlinear parabolic PDEs.For this purpose,we will put the Ef-supermartingales in a Markovian framework.We will make the following assumptions:Let b,h,?:[0,T]× R ? R and f:[0,T]× R3 ? R be deterministic functions and satisfy the following conditions(H4.1)b,h,?,f are continuous in t;(H4.2)There exists a constant L>0,such that|b(t,x)-b(t,x')| + |h(t,x)-h(t,x')|+|?(t,x)-?(t,x')| ? L|x-x'|,|f(t,x,y,z)-f(t,y',z')|? L(|x-x'| + |y-y'| + |z-z'|).For each t ?[0,T]and ??LG2(?t),we consider the following type of SDE driven by 1-dimensional G-Brownian motion:dXst,? = b(s,Xxt,?)ds + h(s,Xxt,?)d<B>s + ?(s,Xst,?)dBs,Xtt,?=?,(3)and the following type of PDE:(?)tu + F(Dx2u,Dxu,u,x,t)= 0,(4)where F(Dx2u,Dxu,u,x,t)=G(H(Dx2u,Dxu,u,x,t))+ b(t,x)Dxu+ f(t,x,u,?(t,x)Dxu),H(Dx2u,Dxu,u,x,t)=?2(t,x)Dx2u + 2h(t,x)Dxu.Theorem 3.Assume(H4.1)and(H4.2)hold.Let u:[0,T]× R ? R be uniformly continuous with respect to(t,x)and satisfy|u(t,x)|?C(1 + |x|k),t ?[0,T],x ?R,where k is a positive integer.Then u is a viscosity supersolution of equation(4),if and only if{Yst,x}s?[t,T]:={u(s,Sst,x)}s?x[t,T]is an Eft,x-supermartingale,for each fixed(t,x)?(0,T)×R,where ft,x,(s,y,z)= f(s,Xxt,x,y,z)and {Xst,x}s?[t,T]is given by(3).Applying the above results,we obtain the following "inverse" comparison the-orem for viscosity solutions of PDEs.Corollary 1.Let V:[0,T]× R ? R be uniformly continuous with respect to(t,x)and satisfy|V(t,x)|?C(1 + |x|k),t?[0,T],x?R,where k is a positive i'n.teger.Assume that V(t,x)? ut1,V(t1,·(t,x),(?)(t,x)?[0,t1]×R,t1?[0,T],where ut1,V(t1,·)denotes the viscosity solution of PDE(4)on(0,t1)× R with Cauchy condition ut1,V(t1,·)(t1,x)=V(t1,x).Then V is a viscosity supersolution of PDE(4)on(0,T)× R.2.1-dimensional Reflected BSDE driven by G-Brownian Mo-tionIn this chapter,we first introduce the definition of solutions to reflected G-BSDEs with a lower obstacle.For simplicity,we only consider the reflected BSDE driven by 1-dimensional G-Brownian motion.We first give the following data:the generators f and g,the obstacle{St}t?[0,T]and the terminal value where the generators are the following mapping f(t,W,y,z),g(t,W,y,z):[0,T]×?T × R2?R.We will make the following assumptions:There exists some ?>2 and L>0 such that(H1)for any y,z,f(·,·,y,z),g{·,·,y,z)G MG?(0,T);(H2)|f(t,y,z)-f(t,y',z')|+ |g(t,y,z)-g(t,y',z')| ? L(|y-y'| + |z-z'|);(H3)? ? LG?(?T)and ? ?>ST,q.s.;(H4)There exists a constant c such that {St}t?[0,T]?SG?(0,T)and St<c,(?)t ?[0,T];(H4'){St}t?[0,T]has the following form St + S0+?0t(s)ds + ?0tl(s)d<B>s + ?0t?·(s)dBs,where {b(t)}t?[0,T],{l(t)}t?[0,T]belong to MG?(0,T)and {?(t)}t?[0,T]belongs to HG?(0,T).Let us now introduce our reflected G-BSDE with a lower obstacle.A triple of processes(Y,Z,A)is called a solution of reflected G-BSDE if for some 1<? ? ?the following properties hold:(a)(Y,Z,A)? SG?(0,T)and Yt?St,0?t?T;(b)Yt = ? + ?tT f(s,Ys,Zs)ds + ?tTg{s,Ys,Zs)d<B>s-ZsdBs +(At-At);(c){-0t(Ys-Ss)dAs}t?[0,T]is a decreasing G-martingale.Here we denote by SG?(0,T)the collection of processes(Y,Z,A)such that Y ?SG?(0,T),Z ?GG?(0,T),A is a continuous nondecreasing process with A0=0 and A ? SG?(0,T).For simplicity,we only consider the case where g?0 and l?0.The results still hold for the other cases.Now we give some a priori estimates for the solution of the reflected G-BSDE from which we can derive the uniqueness.Proposition 1.For i = 1,2,let ?i ?LG?(?T),fi satisfy(HI)and(H2)for some?>2.Assume Yti=?i+?tT fi(s,Ys,Zs)ds-?tT ZsidBs+(ATi-Ati),where Yi ? SG?(0,T),Zi ? HG?(0,T),Ai is a continuous nondecreasing process with A0i = 0 and Ai ? SG?(0,T)for some 1<?<?.Set Yt = Yt1-Yt2,Zt = Zt1-Zt2.Then there exists a con.stant C:= C(?,T,L,?)such that(?)Proposition 2.For i =1,2,let ?i?LG?(?T)with ?i? STi,where(?)and {bi(s)} ? MG?(0,T),{?i(s)}?HG?(0,T)for some ?>2.Let fi satisfy(HI)and(H2).Assume that(Yi,Zi,Ai)?(0,T)for some 1<?<? are the solutions of the reflected G-BSDEs corresponding to ?i,fi and Si.Set Yt =(Yt1-St1)-(Yt2-St2).Then there exists a constant C:= C(?,T,L,?)such that(?)where ?? ?1-ST1-(?2-ST2),?s = |(f1(s,Ys2,Zs2)-f2(s,Ys2,Zs2)|,ps=|b1(s)-b2(s)|+ |?1(s)-?2(s)|,Ss= = Ss1-Ss2 and ?si,0 = |fi(s,0,0)| + |bi(s)| + |?i(s)|.Proposition 3.Let(?1,f1,S1)and(?2,f2,S2)be two sets of data,each one sat-isfying all the assumptions(H1)-(H4).Let(Yi,Zi,Ai)? SG?(0,T)be a solution of the reflected G-BSDE with data(?i,fi,Si),i= 1,2 respectively with 2 ? ?<?.Set Yt=Yt1-Yt2,St=St1-St2,?=?1-?2.Then there exists a constant C:= C(?,T,L,?,c)>0 such that(?)where ?s = |f1(s,Ys2,Zs2)-f2(s,Ys2,Zs2)| and(?)By the above propositions,we can show that the solution of reflected G-BSDE with a lower obstacle is unique.Thanks to[22],we prove the existence of solutions to reflected G-BSDE by approximation method via penalization.Now we consider the following family of G-BSDEs parameterized by n = 1,2,…,(?)By applying G-Ito's formula appropriately,we have the following uniform estimates:there exists a constant C independent of n,such that for 1<?<?(?)By the above analysis and the continuity property of elements in SG?(0,T),we obtain the following convergence result:(?)In summary,the priori estimate and the convergence property of penalized G-BSDEs prove the uniqueness and existence of the solutions to reflected G-BSDEs.Theorem 4.Suppose that ? f satisfy(H1)-(H3)and S satisfies(H4)or(H4')Then the reflected G-BSDE with data(?,f,S)has a unique solution(Y,Z,A).More-over,for any 2? ?<? we have Y?SG?(0,T),Z?HG?(,T)an.d A ?SG?(0,T).If the generator g ? 0,we have a similar result.Theorem 5.Suppose that ?,f and g satisfy(H1)-(H3),S satisfies(H4)or(H4')Then the reflected G-BSDE has a unique solution(Y,Z,A).Moreover,for any 2 ? ?<? we have Y E SG?(0,T),Z ? HG?(0,T)and A ? SG?(0,T).We also have the comparison theorem,similar to that of[34]for non-reflected G-BSDEs.Theorem 6.Let(?1,f1,g1,S1)and(?2,f2,g2,S2)be two sets of data.Suppose Si satisfy(H4)or(H4'),fi and gi satisfy(H1)-(H3)for i = 1,2.In addition,we farther assume the following:(i)?1??2,q.s.;(ii)f1(t,y,x)?f2(t,y,z),g1(t,y,z)?g2(t,y,z),(?)(y,z)?R2;(iii)St1 ?St2,0?t?T,q.s.Let(Yi,Zi,Ai)be a solution of the reflected G-BSDE with data(?i,fi,gi,Si),i = 1,2 respectively.Then Yt1?Yt2,0?t?T q.s.Then we give a probabilistic representation for solutions of some obstacle prob-lems for fully nonlinear parabolic PDEs using the results we obtained in previous sections.For this purpose,we need to put the reflected G-BSDE in a Markovian framework.For each(t,x)?[0,T]×Rd let {Xst,x,t?s?T} be the unique Rd-valued solution of the SDE:(?)We assume that the data(?t,x,ft,x,gt,x,St,x)of the reflected G-BSDE take the following form:?t,x=?(XTt,x),ft,x(s,y,z)=f(s,Xs,t,x,y,z),fijt,x(s,y,z)=gij(s,Ss,t,x,y,z),Sst,x=h(s,Sst,x),where b:[0,T]×Rd?Rd,lij:[0,T]×Rd?Rd,?i:[0,T]×Rd?Rd,?:Rd?R,f,gij:[0,T]× Rd × R × Rd ? R and h:[0,T]× Rn ?R are deterministic functions and satisfy the following conditions:(A1)lij = lji and gij= gji for 1?i,j?d;(A2)b,lij,?i,f,gij,h are continuous in t;(A3)There exist a positive integer m and a constant L such that(A4)h is Lipschitz continuous w.r.t x and bounded from above,h(T,x)??(x)for any x ? Rd;(A4')h belongs to the space CLip1,2([0,T]× Rd)and h(T,x)??(x)for any ? Rd,where CLip1,2[0,T]×Rd)is the space of all functions of class C1,2([0,T]× Rd)whose partial derivatives of order less than or equal to 2 and itself are Lipschtiz continuous functions with respect to x.We now define u(t,x):=Ytt,x,(t,x,)?[0,T]×Rd.It is easy to check that u(t,x)is a deterministic function.Furthermore,we could show that it is continuous with respect to(t,x).Finally,we have the following theorem.Theorem 7.The function u is the unique viscosity solution of the following obstacle problem:where F(Dx2u,Dxu,u,x,t)=G(H(Dx2u,Dxu,Dxu,u,x,t))+(b<t,x>,Dxu>+ f(t,x,u,<?1(t,x),Dxu>,…,<?d(t,x),Dxu>),H(Dx2u,Dxu,u,x,t)?<Dx2u?i(t,x),?j(t,x))+ 2<Dxu,lij(t,x)>+ 2gij(t,x,u<?1(t,x),Dxu>,…,(?d(t,x),Dxu>).At the end of this chapter,we consider the pricing problem for American options under volatility uncertainty in financial market.By applying the results of reflected G-BSDE and the results of stopping time obtained in Hu and Peng[37],Li and Peng[48],we give the superhedging price for this contingent claims.3.Reflected G-BSDE with an Upper ObstacleIn this chapter,we consider the reflected G-BSDE with an upper obstacle.We first give the following data:the generators f and g,the obstacle process {St}tE[0,T]and the terminal value ?,where f and g are maps f(t,w,y,z),g(t,w,y,z):[0,T]×?T×R2 ?R.The following assumptions will be needed throughout this section.There exists some ?>2 and L>0 such that(Al)for any y,z,f(·,·,y,z),g(·,·,y,z)?MG?(0,T)and(?)(A2)|f(t,y,z)-f(t,y',z')| + |g{t,y,z)-g(t,y',z')| ? L(|y-y'| + |z-z'|);(A3){st}t?[0,T]? SG?(0,T)is of the following form St = S0+?0tb(s)+ds+?0tl(s)d+<B>s+?0t?(s)dBs,where {b(t)},{l(t)}?Mg?(0,T),{?(t)}?HG?(0,T)and E[supt?[0,T]{|b(t)|?+|l(t)|?+|?(t)|?}]<?;(A4)??LG?(?T)and ? ?<ST q.s..Then we can introduce our reflected G-BSDE with an upper obstacle.A triple of processes(Y,Z,A)is called a solution of reflected G-BSDE with data(?,f,g,S)if for some 1<? ? ? the following properties are satisfied:(i)(Y,Z,A)? SG?(0,T)and Yt?St,0?t?T;(ii)Yt = ? + ?tTf(s,Ys,Zs)ds + ?tT g(s,Ys,Zs)d<B>s-?tT ZsdBs +(AT-At);(iii){-?0t(Ss-Ys)dAs}t?[0,T]is a decreasing G-martingale.Here we denote by SG?(0,T)the collection of processes(Y,Z,A)such that Y ?SG?(0,T),Z?HG?(0,T),A is a continuous process with finite variation satisfying A0 = 0 and-A is a G-submartingale.For simplicity,we only consider the case where g?0 and l ? 0.We apply the approximation method via penalization to obtain the existence.Now we consider the following family of G-BSDEs parameterized by n = 1,2,…,Ytn = ?+?tTf(s,Ysn,Zsn)ds-n ?tT(Ysn-Ss)+ds-?tT ZsndBs-(KTn-Ktn).Different from the proof for lower obstacle case,we first get the uniform es-timates for the sequence{Yn}n?=1.Then combining this result and the Girsanov transformation imply the following more accurate estimates(?)Then we can derive the uniform estimates for {Ln}n=1? and{Kn}n=1? respectively,where Ltn=-n ?0t(Ysn-Ss)+ds.The remainder proofs are similar to those of lower obstacle case.Since it is difficult to derive some a priori estimates,we cannot prove the uniqueness for solution of reflected G-BSDE with an upper obstacle.However,we establish a variant comparison theorem which yields that the solution we obtained by the above procedure is the largest one.In this sense,we may consider the solution is unique.In conclusion,we have the following theorem.Theorem 8.Under the above assumptions,in particular(A1)-(A4),the reflected G-BSDE with data(?,f,g,S)has a solution(Y,Z,A).This solution is the maximal one in the sense that,if(Y',Z',A')is another solution,then Yt?Y't,for all t ?[0,T].4.Stochastic Optimal Control Problem with Constraints un-der G-frameworkIn this chapter,we explore the stochastic recursive optimal control problem with the obstacle constraints for the cost function under G-framework.This means that the cost function is defined by the solution of the reflected G-BSDE whose solution is required to be above the obstacle.First,we introduce the following notations about admissible controls.Assume U is a given compact subset of Rm.Definition 2.For each given t ?0,u:[t,T]× ?? U is said to be an admissible control on[t,T],if u ?MG2(t,T;T;Rn).The set of admissible controls on[t,T]is denoted by U[t,T].Then,consider the state equation given by the following controlled G-SDE:(?)The cost function is described by the solution Ytt,x,u of the following reflected G-BSDE-dYst,x,u =f(s,Xst,x,u,Yst,x,u,Zst,x,u,us)ds+gij(s,Sst,x,u,Yst,x,u,Zst,x,u,us)d<Bi,Bj>s-Zst,x,udBs+dAst,x,u,YT,t,x,u=?(XT,?,u),Yst,x,u?l(s,Sxt,x,u).Here b,hij = hji:[0,T]× Rn × U ? Rn and a:[0,T]× Rn ×U?Rn×d,?:Rn ? R,f,fij = gij:[0,T]× Rn × R × Rd ×U?R and l:[0,T]×Rn ? R are deterministic continuous functions satisfying the following conditions:(A1)There exists some constant L>0 such that(?)(A2)There exists a constant L>0 such that(?)(A3)There is a constant c such that l ? c and l(T,x)<?(x)for any x ? Rn;(A3')l belongs to the space CLip1,2([0,T]× Rn)and l{T,x)? ?(x)for any x ? Rn,where CLip1,2([0,T]× Rn)is the space of all functions of class C1,2([0,T]×n)whose partial derivatives of order less than or equal to 2 and itself are Lipschtiz continuous functions with respect to x.Then the value function of the stochastic recursive optimal control problem can be defined as the following:(?)We emphasize that the essential infimum of a family of random variables should be defined in the "quasi-surely" sense(q.s.for short).Different from the classical case,this essential infimum may not exist.Motivated by the results in[32],we can prove that V is well-defined and it is a deterministic and continuous function.Then we establish the dynamic programming principle(DPP for short)for the above optimal control problem using the "implied partition" approach in[32].This result extends the one in[32]to the obstacle constraints case for the cost function.We finally proved that the value function is the unique viscosity solution of the obstacle problem for the so-called Hamilton-Jacobi-Bellman-Isaac(HJBI for short)equation.Theorem 9.The value function defined above is the unique viscosity solution of the following HJBI equation:(?)where H and F are given by H(t,x,v,p,A,u)=G(F(t,x,v,p,A,u))+<p,b(t,x,u)>+f(t,x,v,?T(t,x,u)p,u),Fij(t,x,v,p,A,u)=(?T(t,x,u)A?(t,x,u))ij+2<p,hij(t,x,u)>+2gij(t,x,v,?T(t,x,u)p,u),for each(t,x,v,p,A,u)?[0,T]× Rn × R x Rn × S(n)× U.