L~p Solutions For Multidimensional BDSDEs Under The Locally Weak Monotonicity Conditions

This paper studies the existence and uniqueness theorems of L~p(1<p≤2)solutions for multidimensional backward doubly stochastic differential equations(BDSDEs for short)under the globally(locally)monotone condition and p-order globally(locally)weak mono-tonicity condition.Comparison theorems for one-dimensional BDSDEs are also put forward and proved.Assume that generator g satisfies Lipschitz condition with respect to(y,z)in this paper.Chapter 1 introduces the research backgrounds,status and significance,main contribu-tions and some preliminaries of this paper.Chapter 2 proves the existence and uniqueness result and comparison theorem of L~p solutions under the globally monotone condition(see Theorem 2.3 and Theorem 2.4)by virtue of two priori estimates and truncation method.On this basis,proving the existence and uniqueness theorem of L~p solutions under the locally monotone condition(see Theorem2.8)by virtue of the method in Proposition 3.4 of[18].The conclusions of this chapter extends the corresponding results in[39]and[18]to some extent.Chapter 3 assumes that generator f satisfies a special growth condition in(y,z).Firstly,proving the existence and uniqueness result and comparison theorem of L~p solutions under the p-order globally weak monotonicity condition(see Theorem 3.5 and Theorem 3.7)by virtue of the existence result of L~2 solutions and truncation method.On this basis,proving the existence and uniqueness theorem of L~p solutions under the p-order locally weak mono-tonicity condition(see Theorem 3.11)by the similar method in Chapter 2.The conclusions of this chapter extends the corresponding results in[5]and[4]to some extent.Chapter 4 summarizes the results obtained and the methods used in this paper.