L~p Solutions Of Reflected BSDEs With Two Continuous Barriers Under General Assumptions

This paper studies the existence and uniqueness of Lp(p>1)solutions for doubly reflected backward stochastic differential equations(DRBSDEs for short)with two continuous barriers under general assumptions,which strengthens and unifies some existing results.Chapter 1 briefly introduces the research background,research status and the research content.Chapter 2 introduces some notations and definitions of solution for the classic backward stochastic differential equations(BSDEs for short),reflected BSDEs with one continuous barrier(RBSDEs for short)and DRBSDEs.And,some important results about classic BSDEs and RBSDEs are introduced.Chapter 3 first verifies that the generalized Mokobodzki condition is necessary for existence of Lp solutions of DRBSDEs(see Theorem 3.1).Then,under the condition that the generator g satisfies the one-sided Osgood condition with respect to y and is uniformly continuous with respect to z,the comparison theorem for Lp solutions of DRBSDEs is studied(see Theorem 3.3).Finally,the uniqueness theorem of the DRBSDE solution under the above conditions is obtained(see Theorem 3.6).Chapter 4 first studies the convergence of the sequence of Lp solutions for penalized RBSDEs,where the generator g is continuous and has a general growth with respect to(y,z)(see Proposition 4.1).Secondly,under the condition that the generator g satisfies the one-sided Osgood condition,has a general growth with respect to y and is uniformly continuous in z,the uniform estimate on the sequence of Lp solutions of the penalization equations for RBSDEs and non-reflected BSDEs is established(see Proposition 4.2).Finally,based on the above results,a general existence and unique theorem on the Lp solution of DRBSDE is proved(see Theorem 4.4).At the same time,under the condition that the generator g is stronger continuous in(y,z)and has a stronger linear growth in z,a general existence theorem for the Lp solution of DRBSDE is studied(see Theorem 4.6).In Chapter 5,the convergence of the sequence for Lp solutions of DRBSDEs under some elementary conditions is proved(see Proposition 5.1).Furthermore,the existence theorem for the Lp solution of DRBSDE is established,where the generator g is discontinuous in y(see Theorem 5.4).Chapter 4 and Chapter 5,a number of notes and examples are given,and these examples can be applied to the main results of this article,but can not be obtained by any existing results is clarified.The results obtained in this article and the methods used in this article strengthen and unify some existing corresponding results in Fan[16]、FanJiang[19]、Hamadène-Lepeltier-Matoussi[26]、Hamadène-Popier[28]and Klimsiak[37].Chapter 6 briefly summarizes and prospects the main results obtained and the techniques used in this paper.