Study On Solutions Of Backward Stochastic Differential Equations Under General Stochastic Conditions

In this paper,we study the（>1）solutions of backward stochastic differential equations（BSDEs）with generator2)satisfying a-order weak stochastic-monotonicity condition inand the solutions of BSDEs in general space under a stochastic Lipschitz condition.The results improve and generalize some known results.In Chapter 1,we firstly introduce the relevant background and research status of BSDEs,and then explain the research content.Finally,the notations and lemmas needed in this study are introduced to provide preparation for the proof of the main conclusions.In Chapter 2,we research the existence and uniqueness for the（>1）solutions of multidimensional BSDEs where the generator2)satisfies a weak stochastic-monotonicity condition inand a stochastic-Lipschitz condition in.The existence and uniqueness theorem is proved by the convolution technique,a stochastic Gronwall-type inequality,a stochastic Bihari-type inequality and a priori estimates.The above results generalize the corresponding results in Li-Xu-Fan[2019].In Chapter 3,we establish the comparison theorem and the existence and uniqueness for the（>1）solutions of one-dimensional BSDEs with generator2)satisfying a weak stochastic-monotonicity condition inand a stochastic uniform-continuity condition in.The comparison theorem is proved by using the formula of It?o-Tanaka,Girsanov’s trans-formation and the stochastic Bihari-type inequality.We obtain the existence and unique-ness for the solutions of BSDEs by the convolution technique and the results established in Chapter 2.The result generalizes the corresponding result in Wang-Liao-Fan[2019].In Chapter 4,we study the solutions of one-dimensional BSDEs in general space un-der a stochastic-Lipschitz condition with respect to（,）.We use the a prior estimates and the contraction mapping theorem to prove the existence and uniqueness and use the stop-ping time technique,the formula of It?o-Tanaka and Girsanov’s transformation to prove the comparison theorem,which improves corresponding results in Bender-Kohlmann[2000]and Liu-Li-Fan[2020]to some extent.Chapter 5 summarizes the research of this paper and looks forward to the subsequent theoretical research of BSDEs.