A Viscosity Approach to Stochastic Differential Games of Control and Stopping Involving Impulsive Control

TL;DR

This paper studies a stochastic differential game involving impulse control and stopping, establishing the existence of a value function characterized as a unique viscosity solution to a complex HJBI equation.

Contribution

It introduces a viscosity approach to analyze a stochastic game with impulse control and stopping, proving the existence and uniqueness of the value function.

Findings

The value functions are regular and bounded.

The game admits a well-defined value.

The value function solves a double obstacle quasi-integro-variational inequality.

Abstract

This paper analyses a stochastic differential game of control and stopping in which one of the players modifies a diffusion process using impulse controls, an adversary then chooses a stopping time to end the game. The paper firstly establishes the regularity and boundedness of the upper and lower value functions from which an appropriate variant of the dynamic programming principle (DPP) is derived. It is then proven that the upper and lower value functions coincide so that the game admits a value and that the value of the game is a unique viscosity solution to a HJBI equation described by a double obstacle quasi-integro-variational inequality.