# Zero-sum stochastic differential game in finite horizon involving impulse controls

## TL;DR

This paper studies a two-player zero-sum stochastic differential game with impulse controls over a finite horizon, establishing existence, uniqueness, and the equivalence of value functions via viscosity solutions of the HJBI PDE.

## Contribution

It introduces a framework for analyzing such games with weak assumptions on cost functions and proves key properties of the value functions.

## Key findings

- Existence and uniqueness of the solution to the HJBI PDE.
- The upper and lower value functions coincide.
- Validation of the dynamic programming principle for the game.

## Abstract

This paper considers the problem of two-player zero-sum stochastic differential game with both players adopting impulse controls in finite horizon under rather weak assumptions on the cost functions ($c$ and $\chi$ not decreasing in time). We use the dynamic programming principle and viscosity solutions approach to show existence and uniqueness of a solution for the Hamilton-Jacobi-Bellman-Isaacs (HJBI) partial differential equation (PDE) of the game. We prove that the upper and lower value functions coincide.