Solutions For Backward Stochastic Differential Equations With Unbounded Random Coefficients

This paper mainly studies the backward stochastic differential equations(BSDEs for short)with unbounded random coefficients,Two important inequalities,the random Gronwall inequality and the random Bihari inequality,are proposed,and the existence of bounded solutions,the comparison theorem and uniqueness for square growth BSDE with unbounded random coefficients are p roved.To some extent,The existing research results have been extended in this paper.In the first c hapter,t he r esearch b ackground a nd d evelopment s tatus o f B SDE are analyzed firstly,the research content,significance and some preliminary knowledge used in the paper are introduced.In the second chapter,the basic form of stochastic backward Gronwall's inequality(see Theorem 2.3)is put forward and uses respectively the iterative method,the integral method and the martingale representation method to prove it.Then,it presents an application to prove a comparison theorem of solutions for one-dimensional BSDEs under the stochastic Lipschitz condition.This inequality plays an important role in the study of BSDE with unbounded random coefficients.The corresponding results in literatures [46,75,76] are generalized at some extent by the conclusions of this chapter.In the third chapter,the existence,uniqueness and comparison theorem of bounded solutions and maximal and minimal bounded solutions for quadratic BSDEs with unbounded random coefficients are s tudied.Firstly,it proves a class of stochastic Bihari;s inequality(see Proposition 3.1)and priori estimation inequality by using martingale theory,Ito? formual,BMO-martingale theory and Girsanov transformation technique.Secondly,it proves the existence of bounded solutions,maximum and minimum bounded solutions for quadratic BSDEs with unbounded random coefficients(see Theorem 3.10 and 3.12),where the generator 2)has the one-side stochastic super-linear growth in ,and the general quadratic growth in ,and it gives the comparison theorem of maximum and minimum bounded solutions(see Theorem 3.15).Finally,it proves the comparison theorem of bounded solutions,where the generator 2)satisfies the one-side stochastic Osgood condition in ,and stochastic local lipschitz or uniformly continuous or convex or concave condition in (see Theorem 3.17 and 3.18),and it gives the existence and uniqueness of bounded solutions for quadratic BSDEs with unbounded random coefficients by combining the previous existence results(see Theorem 3.19).The corresponding results in literatures [14,17,19,21,22,31] are generelized at some extent by the conclusion of this chapter.In the forth chapter,the results obtained and the methods used in this paper are summarized,And the prospect of the research is given.