Zero-sum Stochastic Differential Games of Impulse Control with Random Intervention Costs

TL;DR

This paper analyzes a finite-horizon zero-sum stochastic differential game with impulse controls, allowing state-dependent intervention costs without monotonicity constraints, and establishes the existence of a game value.

Contribution

It introduces a control randomization approach for the game and proves the value exists even with complex, state-dependent costs and no monotonicity assumptions.

Findings

Existence of a game value for the impulse control game.

Extension to state-dependent intervention costs.

No monotonicity assumptions required on coefficients.

Abstract

We consider a finite-horizon, zero-sum game in which both players control a stochastic differential equation by invoking impulses. We derive a control randomization formulation of the game and use the existence of a value for the randomized game to show that the upper and lower value functions of the original game coincide. The main contribution of the present work is that we can allow intervention costs that are functions of the state as well as time, and that we do not need to impose any monotonicity assumptions on the involved coefficients.