Anticipated Backward Stochastic Differential Equations With Mean Reflection

This paper mainly studies the existence and the uniqueness of the deterministic flat solution(Y,Z,K)to the anticipated backward stochastic differential equation with mean reflection as follows:where the non-decreasing process K is required to be deterministic,and a flat solution means that the solution satisfies ∫0T E[l(t,Yt)]dKt=0.In this paper,the first part consists of the first two chapters,which mainly introduces this new type of backward stochastic differential equation and the definition of the deter-ministic flat solution,the assumptions we need and a priori estimates.The second part is the content of Chapter 3,which mainly introduces the existence and the uniqueness of the solution when the loss function l is linear,that is,l(t,y)＝ y-ut.By using the hypothesis(Hf)and condition(Hδ,ζ)-ii)we can prove the uniqueness of the solution.Fur-thermore with the help of the conclusion that there is a unique deterministic flat solution when generators do not depend on y,z and contraction mapping principle we succeed to prove the existence of a deterministic flat solution of the anticipated backward stochastic differential equation with linear mean reflection.The third part is the content of Chapter 4.Based on the previous three chapters,in this chapter we discuss the existence and the uniqueness of the deterministic flat solution for the general loss function l.In non-linear case,we first construct the following operator:Lt:L2(FT)→[0,∞),Xt→ inf{x≥0:[l{t,x+X)]≥ 0},(?)t ∈[0,T].Then we given the following assumptions:(HL)The operator Lt is Lipschitz continuous in L1-nom,uniformly with respect to t,that is,there exists a constant C>0 such that|Lt(X)-Lt(Y)|≤CE[|X-Y|],0≤t≤T,X,Y∈L2(FT).On this basis,by using the conclusion that there is a unique deterministic solution when the generator does not depend on y,z and applying the contraction mapping principle backward to the small time interval through dividing the interval[0,T],we get the main result,that is,the existence and the uniqueness of the deterministic flat solution of the anticipated backward stochastic differential equation with mean reflection.The fourth part is the content of Chapter 6.Using the similar methods in Chapter 4,we obtain the existence and the uniqueness of the deterministic flat solution of the anticipated backward stochastic differential equation with the risk measurement constraint,p(t,Tt)≤qt,0≤t≤T.