Stochastic differential games involving impulse controls and double-obstacle quasi-variational inequalities

TL;DR

This paper analyzes a two-player stochastic differential game with impulse controls, demonstrating that the game has a well-defined value by proving the unique viscosity solution of the associated double-obstacle quasi-variational inequality.

Contribution

It establishes the existence of a game value by linking the stochastic differential game to a double-obstacle quasi-variational inequality and proving the uniqueness of its viscosity solution.

Findings

Upper and lower value functions coincide

The value functions are the unique viscosity solutions

The game admits a well-defined value

Abstract

We study a two-player zero-sum stochastic differential game with both players adopting impulse controls, on a finite time horizon. The Hamilton-Jacobi-Bellman-Isaacs (HJBI) partial differential equation of the game turns out to be a double-obstacle quasi-variational inequality, therefore the two obstacles are implicitly given. We prove that the upper and lower value functions coincide, indeed we show, by means of the dynamic programming principle for the stochastic differential game, that they are the unique viscosity solution to the HJBI equation, therefore proving that the game admits a value.