The Optimal Objective Problem Of Stochastic Differential Games

In this paper we will consider the optimal objective problem of stochas-tic differential games described by a nonlinear stochastic differential equa-tion. Stochastic differential game could change into a backward stochastic dif-ferential equation(BSDE in short).There has been more than 20 years since BSDE was first proposed in 1990s and has formed a rich and improved system theory which become one of the most popular areas in the research of probabil-ity theory and stochastic analysis.With their pioneering paper of 1989,Fleming and Souganidis were the first to study in a rigorous manner two-player zero-sum stochastic differential games,which has translated former results on differen-tial games from the purely deterministic into the stochastic framework and has given an important impulse for the research in the theory of stochastic differen-tial games.In the paper of 1999,S.Peng and S.Ji introduce a new perturbation method to solve a recursive utility optimization problem under uncertainty. In this paper, we study the existence of saddle point in a stochastic differential game.We can turn a stochastic differential game where the controlled system is described by a nonlinear stochastic differential equation into a stochastic optimal control problem with Girsanov Theorem. We can use the new pertur-bation method to obtain the necessary condition for the existence of optimal objective, which is the necessary condition for the existence of saddle point. This conclusion applies to a special case. We study dynamic measure of risk problem in incomplete market when stock appreciation rates are uncertainty under the constraint of a higher borrowing rate. Applying Ekeland's varia-tional principle, we obtain the form of the optimal objective and the sufficient condition in which the lower-value and the upper-value of the stochastic game are equal,and we also prove the existence of the saddle point.