On The Characters And Applications Of The Solution Of BSDE

The study of Backward Stochastic Differential Equation(BSDE) come from the research of stochastic control and finance etc.; On the contrary, the results of the equation theory applied to the control, finance and PDE etc. fields of math. Relative to Stochastic Differential Equation(SDE),the research of BSDE started later and the results are not so rich. At the same time ,owing to the character of BSDE, it is difficult that stopping time and localization etc. technique be applied to BSDE directly. So the study of BSDE has its own character.So far, BSDEs have 2 types: One is Ito integral type BSDE driven by Brown motion which come directly from stochastic control and it is applied to financial problem later; the other is BSDE with conditional expectation which come directly from financial problem. The conditions of two types of BSDEs are not same, such as the integrability of parameter and coefficient, the filtration etc., the two types of BSDEs are not covered by each other.Relative to SDE, the study for the solution of BSDE under non-Lipschitz condition is absence, especially when the uniqueness of the solution can not be guaranteed, the existence of minimal and maximal solution of BSDE are not be studied.This paper study the character and application of the solution of BSDE, the main results include: for the second kind of BSDE, the existence and uniqueness of the solution under non-Lipschitz condition, comparison theorem and stability are established , under weaker condition , the existence of the minimal and maximal solution is proved and the application in stochastic control and utility function is given; for the first kind of BSDE, under weaker condition , the existence of minimal and maximal solution .stability, comparison theorem and application to utilityfunction are proved.In chapter two, under non-Lipschitz condition, the existence and uniqueness of the solution of the second kind of BSDE is researched, based on it, the stability of thesolution is proved; In chapter three, under non-Lipschitz condition, the comparison theorem of the solution of the second kind of BSDE is proved and using the monotone iterative technique , the existence of minimal and maximal solution is constructively proved; in chapter four, on the base of above results, we get some results of the second kind of BSDE which partly decouple with SDE(FBSDE), which include that the solution of the BSDE is continuous in the initial value of SDE and the application to optimal control and dynamic programming. At the end of this section, the character of the corresponding utility function has been discussed, e.g monotonicity, concavity and risk aversion; in chapter 5, for the first land of BSDE ,using the monotone iterative technique , the existence of minimal and maximal solution is proved and other characters and applications to utility function are studied.