Zero-sum Stochastic Differential Games of Impulse Versus Continuous Control by FBSDEs

TL;DR

This paper studies a stochastic differential game involving impulse and continuous control using forward-backward stochastic differential equations, establishing the dynamic programming principle and proving the existence of a game value through viscosity solutions.

Contribution

It introduces a novel approach to prove the dynamic programming principle for impulse versus continuous control games, handling more general costs and unbounded coefficients.

Findings

DPP established for the game using backward semigroups

Upper and lower value functions are viscosity solutions to the same PDE

Proves the existence of a game value under broad conditions

Abstract

We consider a stochastic differential game in the context of forward-backward stochastic differential equations, where one player implements an impulse control while the opponent controls the system continuously. Utilizing the notion of "backward semigroups" we first prove the dynamic programming principle (DPP) for a truncated version of the problem in a straightforward manner. Relying on a uniform convergence argument then enables us to show the DPP for the general setting. In particular, this avoids technical constraints imposed in previous works dealing with the same problem. Moreover, our approach allows us to consider impulse costs that depend on the present value of the state process in addition to unbounded coefficients. Using the dynamic programming principle we deduce that the upper and lower value functions are both solutions (in viscosity sense) to the same Hamilton-Jacobi-Bellman-Isaacs obstacle problem. By showing uniqueness of solutions to this partial differential inequality we conclude that the game has a value.