Controller-and-Stopper Stochastic Differential Games with Regime Switching

TL;DR

This paper analyzes a stochastic differential game involving a controller and stopper within a regime switching model, establishing the existence of a game value through viscosity solutions of the HJB equation.

Contribution

It introduces a novel approach to proving the existence of a game value in regime switching models using viscosity solutions and dynamic programming principles.

Findings

The lower and upper value functions coincide, confirming the existence of a game value.

Viscosity solutions are used to characterize the value functions.

The approach differs significantly from non-regime switching cases.

Abstract

This paper is concerned with the controller-and-stopper stochastic differential game under a regime switching model in an infinite horizon. The state of the system consists of a number of diffusions \emph{coupled} by a continuous-time finite-state Markov chain. There are two players, one called the controller and the other called the stopper, involved in the game. The goal is to find a saddle point for the two players up to the time that the stopper \emph{terminates} the game. Based on the dynamic programming principle (DPP, for short), the lower and upper value functions are shown to be the viscosity supersolution and viscosity subsolution of the associated Hamilton-Jacobi-Bellman (HJB, for short) equation, respectively. Further, in view of the comparison principle for viscosity solutions, the lower and upper value functions \emph{coincide}, which implies that the game admits a value. All the proofs in this paper are strikingly different from those for the case without regime switching.