The Solutions Of Anticipated Backward Stochastic Differential Equations And Related Problems

In order to solve the problem of general stochastic maximum principle in stochastic control,Professor Peng introduced a new type of equation-non linear backward stochastic differential equation(non linear BSDE for short in the remaining),which was different from the classical stochastic differential equation in the form and methods of solving the problems.In 1990,Professor Pardoux and Professor Peng studied the existence and uniqueness of the adaptive solution,when the generator was Lipschitz and the generator and terminal value was squared integrable.The main result can be seen in Pardoux and Peng(1990).Many researchers have been working on this subject and related properties of the solu-tions of BSDE due to the connection of this subject with mathematical finance,stochastic control,partial differential equations,stochastic games,stochastic geometry and matheinat-ical economics.Moreover,Scholars have studied the Lp(p ? 1)solutions of BSDE and their related problems,and the main result can be seen in Briand et al.(2003).Hu and Peng(2006).In addition,To solve the problem of stochastic control and mathematical finance,In 2009;Peng and Yang have studied the existence and uniqueness of solutions when the gen-erator was Lipschitz and studied the theorem of one-dimensional BSDE.More information about the anticipated BSDE can be found in the literature of Hu and Chen(2006).Yang(2013)and so on.In this paper,we study the existence and uniqueness result of solutions for anticipated BSDE.and establish the corresponding one-dimensional comparison theorems for the type of anticipated BSDE.and give the generalized(g,?)-expectations and related properties.In chapter 1.we briefly introduce the backgrounds and some useful perliminaries,the latest status and the content of research:In chapter 2,we prove the existence and uniqueness re-sult for Lp(p>1)solutions of anticipated BSDE under monotonicity and general increasing conditions on y.with Lipschitz on z,which are both non-uniformly with respect to t.In chapter 3,we put forward and prove the corresponding comparison theorems of the kind of one-dimeusional anticipated BSDE by Tanaka formula and we give the definition of gener-alized(g,?)-expectation.generalized conditional(g,?)and introduce the properities via the solutions of the corresponding BSDE.