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

## TL;DR

This paper develops a BSDE-based framework for zero-sum stochastic differential games with impulse controls, proving the existence of a game value through viscosity solutions of HJBI equations.

## Contribution

It introduces a novel BSDE approach to analyze stochastic differential games with impulse controls, establishing the dynamic programming principle and uniqueness of solutions.

## Key findings

- Proved the dynamic programming principle for the game.
- Established the upper and lower value functions as unique viscosity solutions.
- Showed the game admits a value by proving the coincidence of value functions.

## Abstract

This paper focuses on zero-sum stochastic differential games in the framework of forward-backward 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 \textit{% backward semigroups}, we 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.