Some Results About Anticipated BSDE And Anticipated BSDEwMS

The linear Backward Stochastic Differential Equation (BSDE for short) was first introduced by Bismut (1973) [10]. Pardoux和Peng (1990) [5] proved the existence and uniqueness theorem of the solution of nonlinear BSDE under Lipschitz condition. Duffe and Epstein (1992) [3] also proposed a type of BSDE independently to character-ize the stochastic differential utility. The theory of BSDEs has been studied with great interest in the last less than twenty years because of its connections with stochastic control, partial equation (PDE), mathematical finance and economics.BSDE has great connections with nonlinear partial differential equations (see Bar-les and Lesigne [7], Briand [2], Pardoux [4], Peng [6], etc.) and can be used in non-linear semi-groups, and stochastic control problems (see Quenez [16], El Karoui,Peng and Quenez [12], Hamadene and Lepeltier [24] and Peng [19]). At the same time, in mathematical finance, the theory of the hedging and pricing of a contingent claim is typically expressed in terms of a linear BSDE (see El Karoui, Peng and Quenez [12]). In 1997 Peng [20] introduced a kind of nonlinear expectation:g-expectation via a par-ticular BSDE. Using Peng's g-expectation, it is easy to define conditional expectation. Rosazza [8] considered a type of dynamic risk measures via g-expectations. Peng [18] defined filtration consistent evaluation satisfying some restrictions is a g-evaluation, that is, whatever model or mechanism used to evaluate, once it satisfies the restrictions, there is a BSDE behind of it, the generator g is its mechanism, and the solution of BSDE is the evaluation.On the other hand, the comparison theorems of two stochastic differential equa-tions have received a lot of attention (see Anderson [27], Ikeda and Watanable [17], Mao [28], Skorohod [1], Yamada [25], Yan [29], Peng and Zhu [23]). Based on the above results, Yang [22] researched a new type of equations:anticipated backward stochastic differential equations (anticipated BSDEs for short). There exists perfect du-ality between them and stochastic differential delay equations (SDDEs for short), and proved the existence and uniqueness theorem of the solutions of anticipated BSDEs and anticipated BSDEs with stopping times, the comparison theorem of the solutions of anticipated BSDEs and SDDEwMS (see Chenggui Yuan,Zhe Yang and Xuerong Mao [15]). In this paper, we prove the existence and uniqueness theorem and compar-ison theorem of the solutions of BSDEwMS and anticipated BSDEwMS. At the same time, we prove the existence and uniqueness theorem of anticipated BSDEs with con-tinuous coefficient and non-Lipschitz conditions, and prove the stability of the solutions of anticipated BSDEs.The paper is organized as follows:Chapter 1 mainly introduces the basic concepts and lemmas. Chapter 2 proves the stability of the solutions of anticipated BSDEs, and the existence and uniqueness theorem of anticipated BSDEs with continuous coefficient and non-Lipschitz conditions. The following are the main results of Chapter 2.Theorem 2.1:(Stability Theorem) If anticipated BSDE satisfies (H3), (H4) and (H5), we haveTheorem 2.2:(Existence Theorem of Adapted Solution) if f satisfies: (H6) linear growth: (H7) for fixed s,ω, y, z, f (s,ω,·,·,·) is continuous and f(s,ω,y,z,·) is increas-ing. Then, if the following anticipated BSDE has adapted solution, i.e., (?)-adapted process satisfies (2.3). Theorem 2.3:Denote(-Yi,-Zi), i= 1,2, are the minimal solutions of the following equation, where fi satisfies (H1) and(H2), for further, if are the maximal solutions of (2.6), thenTheorem 2.4:(Existence and Uniqueness of the Adapted Solution Theorem) If f satisfies (H2) and (H8),δ,ζsatisfy (1) and (2), then for any given terminal condition the anticipated BSDE (1.3) has a unique solution.Chapter 3 studies the existence and uniqueness theorem and comparison theorem of the solutions of BSDEwMS. The following are the main results of Chapter 3.Theorem 3.1:(Existence and Uniqueness of the Adapted Solution Theorem) If f satisfies (H9) and (H10), then for any given terminal condition (3.1) has a unique solution, i.e., there exists a unique(?) -adapted process (Y., Z.)∈satisfies (3.1).Theorem 3.2:(Comparison Theorem) If R satisfies (H9) and (H10), j= 1,2, let (Y(j)) and Z(j)) denote the solutions of the following BSDEwMS separately: and we have the strict comparison theorem:under the above hypothesis,Chapter 4 studies the existence and uniqueness theorem and comparison theorem of the solutions of anticipated BSDEwMS. The following are the main results of Chap-ter4.Theorem 4.1:(Existence and Uniqueness of the Adapted Solution Theorem) If f satisfies (H1)'and (H2)',δ,ζsatisfy (1) and (2). Then for any given terminal condition has a unique solution.Theorem 4.2:(Comparison Theorem) If fj satisfies (H1)'and (H2)',j = 1,2,δsatisfies (1) and (2), and is increasing, i.e., if have denote the solutions of the fol-lowing anticipated BSDEwMS separately: and we have the strict comparison theorem:under the above hypothesis,...