### A BSDE Approach to Stochastic Differential Games Involving Impulse Controls and HJBI Equation

ZHANG Liangquan

1. School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China
• Received:2020-10-26 Revised:2021-03-27 Published:2022-06-20
• Supported by:
This research was supported in part by the National Natural Science Foundation of China under Grant Nos. 62173255 and 62188101.

ZHANG Liangquan. A BSDE Approach to Stochastic Differential Games Involving Impulse Controls and HJBI Equation[J]. Journal of Systems Science and Complexity, 2022, 35(3): 766-801.

This paper focuses on zero-sum stochastic differential games in the framework of forwardbackward stochastic differential equations on a finite time horizon with both players adopting impulse controls. By means of BSDE methods, in particular that of the notion from Peng's stochastic backward semigroups, the authors prove a dynamic programming principle for both the upper and the lower value functions of the game. The upper and the lower value functions are then shown to be the unique viscosity solutions of the Hamilton-Jacobi-Bellman-Isaacs equations with a double-obstacle. As a consequence, the uniqueness implies that the upper and lower value functions coincide and the game admits a value.
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