Backward Stochastic Differential Equations And Financial Applications

In 1973,Bismut introduced the stochastic stochastic linear differential equation for the first time to study the stochastic optimal control problem.Since 1990,Pardoux and Peng have been generalized the backward stochastic differential equation which was proposed by Bismut and have made a large number of pioneering works in theory and practice.In the course of the study,mathematicians and economists have found that the backward stochastic differential equation has important theoretical and practical value in random analysis,PDE,financial mathematics and so on,and has done a lot of intensive research.Now the backward stochastic differential equation has become an important branch of stochastic analysis,finance,control and other fields.In finance,some problems of pricing and hedging contingent claims can eventually be transformed into a series of backward stochastic differential equations?This paper is mainly concerned with backward stochastic differential equation(B-SDE)and its application to the finance,the outline of the paper is as follows.In chapter 2 mainly present some examples of BSDEs which appear naturally in the problem of pricing and hedging contingent claims.Besides,another example is given for introducing the definition of recursive utility.In chapter 3 will present some important results about BSDEs.Such as a priori estimates of the different between two solutions,the existence and uniqueness theorems which was proposed by Pardoux and Peng,the comparison theorem,and the properties of continuity and differentiability of the solutions of BSEDs so on.In chapter 4,we study some properties of the BSDE with respect to the generator that are concave function.The paper has proved that this kind of BSDEs can be written in the form of stochastic control problem.Then,we give the application of the solution of the nonlinear backward stochastic differential equation in the financial field,such as application to recursive utility and to Europen option pricing in the constrained case.