A Study Of Solvability Of Backward Stochastic Differential Equations

In[75],Pardoux and Pong introduced the following nonlinear backward stochastic differential equations(BSDEs for short) (?) and proved the existence and uniqueness of the solution to Eq.(2)under the Lipschitz assumption of the generator f.Since then,BSDEs have gradually become an important mathematical tool in many fields such as financial mathematics,stochastic games and optimal control,partial differential equations and so on.For these reasons,many efforts have been made to relax the Lipschitz conditions on the generator f.In this thesis we will mainly focus on the solvability of Eq.(2)with non-Lipschitzian coefficients.In Chapter 1,we study a class of multidimensional BSDEs with uniformly coefficients and show that the corresponding BSDE has a unique solution.In order to get our results,we first construct a sequence of Lipscitz continuous functions(fn)n?0 that,converges to f uniformly on the whole space of(y,z)and let(yn,zn)n?0 be the solutions to Eq.(2)associated with(fn,?)n?0.Secondly,we construct an ODE which solution can dominated the |ym-yn|,m ? n,then we show that the sequence(yn,zn)n?0 will converge to the solution of BSDE(2).For the uniqueness of the solution,we take the similar procedure.And in Chapter 2,we study a class of multidimensional BSDEs with linear growth coefficients.By introducing a new notion of envelope of stochastic process,we construct a family of random differential equations which can be viewed as a family of BSDEs and show that the sequence(yn,zn)n?0 will converge to the solution of BSDE(2).Similar-ly,we prove the solution of BSDE(2)is also unique.In Chapter 3,depending on the method developed in Chapter 2,for a class of BSDEs with quadratic growth,we show the uniqueness of solution of BSDE(2).In Chapter 4,we will prove the double barrier reflected BSDEs have unique solu-tions under the general conditions via penalization method.In order to solve the double barrier reflected BSDEs,the previous methods are essentially to treat one reflecting bar-rier via penalization method by means of the results of one single reflected BSDEs,that is to regard the double barrier reflecting BSDEs as one barrier reflecting BSDEs.In this chapter,with the help of the notion of local solutions introduced in[43],we prove that the BSDE with two reflecting barriers has a unique square-integrable solution via penal-ization method under a sufficient and necessary condition,and give its application to the solvability of decoupled forward-backward SDEs with linear growth rate.Without using the results of one single reflected BSDEs,we treat the double barrier reflected BSDEs by applying directly penalization technique to two reflecting barriers,which means that the double barrier reflected BSDEs can be viewed as BSDEs without reflecting barriers.This thesis consists of five chapters.In the following,we list the main results of this thesis.Chapter 1:In this chapter,we study the solvability of the multidimensional BSDEs with uniformly continuous coefficients.By using some new techniques and methods,and under the following assumptions:(H1.1).The process f(·,·,0,0),(0 ? t ? 1)belongs to H2,d and,for any(y,z)?Rd × Rd×m,the process(f(·,·,y,z))0?t?1 is P-measurable.(H1.2).(i)There exists a continuous non-decreasing function ? from R+ to R+with at most linear growth and satisfying ?(0)= 0 and ?(x)>0 for all x>0 such that:|f(t,y1,z)-f(t,y2,z)| ? ?(|y1-y2|),a.s.,(?)t,y1,y2,z,moreover,?0+[?(x)]-1dx = +?.(ii)There exists a continuous function ? from R+ into itself with at most linear growth and satisfying ?(0)= 0 and ? is a convex function in a neighbourhood of 0 such that:|f(t,y,z1)-f(t,y,z2)|??(?z1-z2?),a.s.,(?)t,y,Z1,z2,We can obtain:Theorem 0.1.If ? ?L2(?,F,P)and f satisfies Assumptions(H1.1)and(H1.2),then there exists a unique pair of processes(y,z)in S2,d × H2,d×m such thatChapter 2:In this Chapter,for a class of the multidimensional BSDEs with linear growth coefficients,under the following assumptions:(H2.1).The process f(·,·,0,0),(0 ? t ? 1)belongs to H2,d and,for any(y,z)?Rd × Rd×m,the process(f(·,·,y,z))0? t ?1 is P-measurable.(H2.2).f is linear growth with respect to(y,z),i.e.,there exists a nonnegative constant K such that |f(t,w,y,z)| ? K(1 + |y| + |z|),(?)(t,?).(H2.3).For all i ?{1,2,…,d},.fi.(t,y,z)=fi(t,y,zi),i.e.,.fi only depends on the ith row of z.(H2.4).For P-a.s.? ? ?,there exists a nondecreasing linear growth function?:R+ ? R+ satisfying ?(0)= 0 and ?(x)>0,(?)x ?(0,+?)such that we show thatTheorem 0.2.If ? ?L2(?,F,P)and f satisfies Assumptions(H2.1)-(H2.4),then there exists a unique pair of processes(y,z)in S2,d × H2,d×m such,thatChapter 3:In this Chapter,for a class of the multidimensional BSDEs with quadratic growth coefficients,under the following assumptions:(H3.1).The process(f·,·,0,0),(0 ? t ? 1)belongs to H2,d and,for any(y,z)?Rd × Rm×d,the process(f·,·,y,z),(0 ? t ? 1 is P-measurable.(H3.2).For all i ? {1,2,…,d},fi(t,y,z)=fi(t,y,zi),i.e.,fi only depends on the ith row of z.(H3.3).For P-a.s.? ? ?,there exists a nondecreasing linear growth function?:R+ ? R+ satisfying ?(0)= 0 and ?(x)>0,(?)x ?(0,+ ?)such that(H3.4).there exists a nonnegative constant K such that,(?)i ? {1,2,…,d},we show thatTheorem 0.3.Suppose ? ? L2(?,F,P)and f satisfies Assumptions(H3.1)-(H3.4),if(g1,z1)and(y2,z2)are two soluttions of the following BSDEChapter 4:In this chapter,we study the solvability of the double barrier reflect-ed BSDEs via penalization method and give its application to the decoupled forward-backward SDEs with linear growth rate.(H4.1)Suppose L<H<U,a.s.,where H is a continuous semimartingale H(t)=R(t)+ M(t)= R(t)+ ?t0 + h(s)dBs,t ?[0,T]satisfying the property:the total variation process of R denoted by S satisfiesThen we haveTheorem 0.4.If g is Lipschitz with respect to(y,z)and the double reflected barriers L,U satisfying L<U,L(T)? ? ? U(T),a.s,and the assumption(H4.1),then there exists a unique quadruple P-measurable processes(y,z,K+,K-)? L2(1;0,T)× L2(1 ×Theorem 0.5.If the driver g(s,x,y,z)in the reflected BSDE(4.29)is Lipschitz in y and z,uniformly over all s ?[0,T]and all x ? Rl,and the assumptio'n(H3.2)holds.Then there exist measurable deterministic functions ?:[0,T]× Rl ? Rm,and?:[0,T]×Rl ? Rm×d,such,that for any 0 ? t ? s ?T,Yt,x(s)= ?(s,Xt,x(s)),Zt,x(s)=?(s,Xt,x(s)).Theorem 0.6.Under the assumption(H4.2),there exists a solution(Y,Z,K+,K-)to the BSDE(4.29)with two reflecting barriers.