RBSDEs With General Time Interval And Generators Satisfying Some Stochasticity Conditions

In this paper,we mainly establish the existence and uniqueness of solutions for backward stochastic differential equations(BSDEs for short)and reflected backward stochastic differential equations(RBSDEs)with continuous obstacle under general time terminal,where the generators satisfy some stochasticity conditions non-uniform in both t and ω.In this paper,we generalize and improve some existing results.In chapter 1,we introduce the background of BSDEs and RBSDEs,the research contents and significance of this paper.In chapter 2,we mainly introduce some useful preliminaries,main hypotheses and related lemmas,which are used to prepare for the proof of main conclusions.In Chapter 3,we prove the existence and uniqueness for solutions of one-dimensional BSDEs with general time terminal where the generator g satisfies the general growth condition and the one-side Osgood condition non-uniform in both t and ω in y and the uniform continuity condition non-uniform in both t and ω in z.Firstly,comparison theorem(see Theorem 3.1)for solutions of one-dimensional BSDEs with general time terminal is proved by using the formula of Ito-Tanaka,Girsanov’s transformation and stochastic Bihari’s inequality.Secondly,we establish the existence and uniqueness of BSDEs solution(see Theorem 3.6)with the help of theorem 3.1,The above results generalize the corresponding results in Li-Xu-Fan[2019].In Chapter 4,we prove the comparison theorem of the solution of one-dimensional RBSDEs with general time terminal and continuous obstacles(see Theorem 4.4),and then use the comparison theorem to obtain the uniqueness of solution of RBSDEs(see Theorem 4.5)where the generator g satisfies the same stochastic conditions as above.Finally,we obtain the existence result of the solution of the equation by using the method of aprior estimations and penalty(see Theorem 4.5).To some extent,the above result generalizes the corresponding result in Fan[2017].In Chapter 5,we summarize the results obtained in this paper,and gives the future research prospect of BSDE and RBSDE theory which we will study.