Anticipated backward stochastic differential equations and their applications to zero-sum stochastic differential games

Abstract

In this article, we are concerned with the zero-sum stochastic differential game problems with the cost functional depending not only on $g(X_T)$ but also on $\{g(X_t),t\in [T,T+\theta ]\}.$ The main approach is the anticipated backward stochastic differential equations under weaker conditions. For these equations, we prove the existence and uniqueness theorem and the comparison theorems. Then, by using these techniques, we get the saddle-point strategy for the zero-sum stochastic differential games when the Isaacs’ condition holds.

Keywords:

• Anticipated backward stochastic differential equation
• Comparison theorem
• Saddle-point
• Zero-sum stochastic differential game

Additional information

Funding

This work is supported by the Program of Natural Science Research of Jiangsu Higher Education Institutions of China (Grant No. 17KJB110009).