Nonzero-sum stochastic differential games with impulse controls: a verification theorem with applications

TL;DR

This paper develops a verification theorem for nonzero-sum impulse stochastic differential games, providing a framework to determine optimal strategies and value functions, with explicit solutions for specific cases and numerical analysis for others.

Contribution

It introduces a verification theorem linking quasi-variational inequalities to game solutions, with explicit Nash equilibria in certain impulse games.

Findings

Explicit Nash equilibrium characterized for a symmetric impulse game.

Derived formulas for optimal strategies and value functions.

Asymptotic analysis of intervention costs and numerical solutions for non-symmetric cases.

Abstract

We consider a general nonzero-sum impulse game with two players. The main mathematical contribution of the paper is a verification theorem which provides, under some regularity conditions, a suitable system of quasi-variational inequalities for the value functions and the optimal strategies of the two players. As an application, we study an impulse game with a one-dimensional state variable, following a real-valued scaled Brownian motion, and two players with linear and symmetric running payoffs. We fully characterize a Nash equilibrium and provide explicit expressions for the optimal strategies and the value functions. We also prove some asymptotic results with respect to the intervention costs. Finally, we consider two further non-symmetric examples where a Nash equilibrium is found numerically.