Existence And Uniqueness Of Solutions And Comparison Theorems For Anticipated Backward Stochastic Differential Equations

In this paper, we study the existence and uniqueness result of square integrable solutions for multidimensional anticipated backward stochastic differential equations (anticipated BSDEs for short in the remaining), and establish the corresponding one-dimensional comparison theorems for the type of anticipated BSDEs. Our results im-prove some known results.In Chapter 1, we briefly introduce the backgrounds, the latest status and the content of research, and we also give some useful preliminaries.In Chapter 2, we study a class of multidimensional anticipated backward stochas-tic differential equations. When the generators/satisfy the non-Lipschitz condition depicted by the kind of special concave function with respect to Y and the anticipated term of Y, and satisfy the Lipschitz condition with respect to Z and the anticipated term of Z which is not uniform on t. We first establish the existence and uniqueness result of square integrable solutions for anticipated BSDEs by Picard-type iteration (see Theorem 2.1). Then, we introduce an exanple to show that Theorem 2.1 generalizes the corresponding results in Peng-Yang [2009], Yang-Robert [2013a], Wu-Wang [2012]. Finally, we put forward and prove the corresponding comparison theorems of the kind of one-dimensional anticipated BSDEs (see Theorem 2.2, Theorem 2.3, Theorem 2.4, Theorem 2.5). We mention that our results also generalize some corresponding results in several existing works such as Peng-Yang [2009], Xu [2011], Wu-Wang [2012] and Zhang [2014].In Chapter 3, we study a class of multidimensional anticipated backward stochas-tic differential equations. When the generators f satisfy the Osgood condition with respect to Y and the anticipated term of Y, and satisfy the Lipschitz condition with re-spect to Z and the anticipated term of Z, we obtain an existence and uniqueness result of square integrable solutions for this kind of anticipated BSDEs by construct the se-quence of uniformly continuous functions, which the sequence is uniformly Lipschitz continuous (see Theorem 3.1).Finally, a summary and prospect of this paper are given in Chapter 4.