Bounded Solutions For General Time Interval Quadratic BSDEs With Stochastic Conditions

This paper mainly studies the stability,existence,comparison and uniqueness of bounded solutions for general time interval one-dimensional quadratic backward stochas-tic differential equations(BSDEs for short)with stochastic conditions,which improve some results of existing works at some extent.Chapter 1 briefly introduces the research backgrounds and status,research contents and significance,and some preliminaries.Chapter 2 firstly proves the monotonic stability of bounded solutions for general time interval one-dimensional quadratic BSDEs with stochastic conditions(see Theorem 2.1)in three steps by virtue of Riesz theorem,Holder inequality,and repeatedly using Ito formula,exponential transformation,basic inequality,Lebesgue dominated convergence theorem tools.Then in view of Tanaka formula,BMO-martingale and Girsanov trans-lation tools,this paper obtains a comparison theorem(see Theorem 2.5)for this kind of BSDEs,where the generator g satisfies a stochastic condition in y,which is not uniform with respect to both ? and t,and a quadratic growth condition in z.This result extends the corresponding results in Briand-Hu[2008]and Fan[2016b].Chapter 3 proves the existence of bounded solutions for general time interval one-dimensional quadratic growth BSDEs(see Theorem 3.3)by virtue of the monotonic sta-bility result and comparison theorem obtained in the previous chapter,in view of BMO-martingale,convolution,stopping time tools and Ito formula,and Fatou lemma,where the generator g has a one-side stochastic linear growth and a general growth in g,and a gener-al quadratic growth in z.Then this paper proves the existence of the minimal or maximal bounded solution for this kind of BSDEs(see Theorem 3.9),where the generator g has a stochastic linear growth in g,and a semi stochastic linear growth and semi-quadratic growth in z.The conclusion of this chapter extends to some extent the corresponding results in Kobylanski[2000],Lepeltier-San Martin[1998],Briand-Hu[2008]and Fan[2016b].Chapter 4 proves the existence of the minimal and maximal bounded solution of gen-eral time interval one-dimensional quadratic growth BSDEs with stochastic conditions(see Theorem 4.2)by deducing two priori estimates of both L2 solutions and bound-ed solutions,constructing convolution,in view of Ito formula,Tanaka formula,BMO-martingale and Girsanov translation tools.Further this paper deduces the comparison theorem of the minimal and maximal bounded solutions(see Theorem 4.4),where the generator g has a one-side stochastic linear growth and a stochastic general growth in y,and a general quadratic growth in z.In addition,all conclusions obtained in this chapter hold still true when T is an(Ft)-stopping time.The conclusion of this chapter extends to some extent the corresponding results in Lepeltier-San Martin[1998]and Fan[2016b].Chapter 5 summarizes the results obtained and the methods used in this paper,and gives the future research prospect of BSDE theory.