On the Multi-Dimensional Controller and Stopper Games

TL;DR

This paper analyzes a complex multi-dimensional stochastic game involving control and stopping, establishing the existence of a value and characterizing it as a viscosity solution to a related PDE.

Contribution

It introduces a framework for multi-dimensional controller-and-stopper games with control over both drift and volatility, proving the existence of a game value and its PDE characterization.

Findings

The game has a well-defined value under certain conditions.

The value function uniquely solves an obstacle Hamilton-Jacobi-Bellman equation.

The model extends previous one-dimensional results to multi-dimensional settings.

Abstract

We consider a zero-sum stochastic differential controller-and-stopper game in which the state process is a controlled diffusion evolving in a multi-dimensional Euclidean space. In this game, the controller affects both the drift and the volatility terms of the state process. Under appropriate conditions, we show that the game has a value and the value function is the unique viscosity solution to an obstacle problem for a Hamilton-Jacobi-Bellman equation.