Study On L~1 Solutions Of BSDEs Under A Stochastic One-Sided Osgood Condition

The thesis establishes existence,uniqueness and stability for the integrable solu-tion(the L~1solution)of multidimensional backward stochastic differential equations(BSDEs)on a general time interval,where the generator g satisfies a stochastic one-sided Osgood condition in y and a stochastic Lipschitz condition in z.The usage of the stochastic Gronwall-type inequality,the stochastic Bihari-type inequality and some methods dealing with stochastic coefficients are both innovative,and may be useful in further study on the integrable solutions.Chapter 1 contains the background of BSDEs,the research contents as well as the significance of this thesis.Chapter 2 mainly introduces some notations,a priori estimates,the stochastic Gronwall-type inequality and the stochastic Bihari-type inequality that play a key role in the proof.In Chapter 3,an existence and uniqueness result for the L~1solution of BSDEs where the generator g satisfies a stochastic one-sided Osgood condition in y,a stochas-tic Lipschitz condition as well as a stochastic sublinear growth condition in z is es-tablished(see Theorem 3.4).To overcome some core difficulties,the chapter uses the stochastic Gronwall-type inequality and the stochastic Bihari-type inequality,obtains existence and uniqueness of the L~1solution.Since the assumptions in Briand-Delyon-Hu-Pardoux-Stoica[2003],Xiao-Fan-Xu[2015]and Fan[2018]are its special case(see Remark 3.5),the result in this chapter extend some existing corresponding results above.In Chapter 4,a stability result for the L~1solutions of BSDEs is established(see Theorem 4.4).This chapter improves the processes in Fan[2018]by strengthening the stochastic sublinear growth condition of the generator in z to a stochastic boundability condition,and obtains the stability result.The chapter generalize the corresponding results in Fan[2018]to some extent.In Chapter 5,we summarize and outlook the results obtained in this thesis.There are totally 86 references in this thesis.