L~1 Solutions Of Multidimensional BSDEs With Time-Varying Uniform Continuity Generators In Z

In this thesis,we establish existence,uniqueness and stability theorem for the L~1solution of multidimensional backward stochastic differential equations under a general time interval by virtue of the convolution technique,Girsanov’s transform and Bihari’s inequality,where the generator g satisfies a time-varying one-sided Osgood condition in y,a time-varying Lipschitz continuous condition and a time-varying uniformly con-tinuous condition in z.This improves some existing results.Chapter 1,we briefly introduce the development background and current situation of backward stochastic differential equations,describe the main content and signifi-cance of this thesis,at the same time,the notations and a priori estimates used in this thesis are sorted out.In Chapter 2,we are devoted to proving existence and uniqueness for the L~1solu-tion of multidimensional backward stochastic differential equations under general time intervals(see Theorem 2.2 and Theorem 2.3),where the generator g satisfies a time-varying one-sided Osgood condition in y and a time-varying Lipschitz condition in z.This chapter overcomes some difficulties caused by general time intervals and estab-lishes existence and uniqueness of the L~1solution by virtue of a priori estimates(see Lemma 1.1-Lemma 1.3)together with various methods such as the Picard iteration.The results in this chapter extend and improve the corresponding results in the litera-tures such as Fan[2018],etc.In Chapter 3,we prove existence,uniqueness and stability theorem for the L~1so-lution of multidimensional backward stochastic differential equations under a general time interval(see Theorem 3.2 and 3.3),where the generator g satisfies a time-varying one-sided Osgood condition in y,a time-varying uniformly continuous condition in z,and the ith component of the generator g only depends on the ith row of the matrix z.Based on Chapter 2,by virtue of the convolution technique to construct a generator sequence g~nwhich uniformly converges to g(see Proposition 3.5),existence of the so-lution is proved in Chapter 3 by applying Lebesgue’s dominated convergence theorem.Then the content of this chapter generalizes proposition 4.1 in Xiao-Fan[2020](see proposition 3.7).By combining Girsanov’s theorem and H?lder’s inequality,we ob-tain uniqueness of the solution,and further prove the stability of the L~1solution.The results in this chapter extend and improve to some extent the corresponding results in the literatures such as Ma-Fan-Fang[2016],Xiao-Fan[2020],Fan[2018],Dong-Fan[2018b],etc.In Chapter 4,we summarize and outlook the results obtained in this thesis.There are totally 92 references in this thesis.