A zero-sum stochastic differential game with impulses, precommitment, and unrestricted cost functions

TL;DR

This paper analyzes a zero-sum stochastic differential game with impulse and stochastic controls, introducing a novel approach that avoids time-decreasing impulse costs by restricting impulses to rational times, and proves the existence of a game value and unique solutions.

Contribution

It establishes the existence of a game value and unique viscosity solutions for a new class of SDGs with impulses and unrestricted costs, using rational times for impulses.

Findings

Game admits a well-defined value.

Existence and uniqueness of viscosity solutions proved.

Impulses restricted to rational times ensure continuity.

Abstract

We study a zero-sum stochastic differential game (SDG) in which one controller plays an impulse control while their opponent plays a stochastic control. We consider an asymmetric setting in which the impulse player commits to, at the start of the game, performing less than q impulses (q can be chosen arbitrarily large). In order to obtain the uniform continuity of the value functions, previous works involving SDGs with impulses assume the cost of an impulse to be decreasing in time. Our work avoids such restrictions by requiring impulses to occur at rational times. We establish that the resulting game admits a value, and in turn, the existence and uniqueness of viscosity solutions to an associated Hamilton-Jacobi-Bellman-Isaacs quasi-variational inequality.