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

doi:10.1007/s11424-021-0264-4

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.

Key words: Dynamic programming principle (DPP), forward-backward stochastic differential equations (FBSDEs), Hamilton-Jacobi-Bellman-Isaacs (HJBI), impulse control, stochastic differential games, value function, viscosity solution

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