On zero-sum Stochastic Differential Games with Jump-Diffusion driven state: A viscosity solution framework

TL;DR

This paper develops a viscosity solution framework for zero-sum stochastic differential games with jump-diffusion driven states, establishing existence, uniqueness, and verification of the game value via nonlinear integro-PDEs.

Contribution

It introduces a novel viscosity solution approach for zero-sum stochastic differential games with jump processes, including a verification theorem for nonlocal equations.

Findings

Existence and uniqueness of the game value as a viscosity solution.

Formulation of a verification theorem for the game.

Application of viscosity solutions to nonlocal integro-PDEs.

Abstract

A zero-sum differential game with controlled jump-diffusion driven state is considered, and studied using a combination of dynamic programming and viscosity solution techniques. We prove, under certain conditions, that the value of the game exists and is the unique viscosity solution of a fully nonlinear integro-partial differential equation. In addition, we formulate and prove a verification theorem for such games within the viscosity solution framework for nonlocal equations