A Zero-Sum Deterministic Impulse Controls Game in Infinite Horizon with a New HJBI QVI

TL;DR

This paper analyzes a two-player zero-sum deterministic impulse control game over an infinite horizon, establishing the existence and uniqueness of the value function as a viscosity solution to a new HJBI QVI.

Contribution

It introduces a novel HJBI QVI framework for impulse control games and proves the coincidence and uniqueness of the value functions under weak assumptions.

Findings

Value functions are continuous and viscosity solutions to the HJBI QVI.

Under a proportional property assumption, the value functions are unique.

The lower and upper value functions coincide, confirming the game's value.

Abstract

In the present paper, we study a two-player zero-sum deterministic differential game with both players adopting impulse controls, in infinite time horizon, under rather weak assumptions on the cost functions. We prove by means of the dynamic programming principle (DPP) that the lower and upper value functions are continuous and viscosity solutions to the corresponding Hamilton-Jacobi-Bellman-Isaacs (HJBI) quasi-variational inequality (QVI). We define a new HJBI QVI for which, under a proportional property assumption on the maximizer cost, the value functions are the unique viscosity solution. We then prove that the lower and upper value functions coincide.