On the Multi-Dimensional Controller and Stopper Games

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.

Key words: Controller-stopper games, weak dynamic programming principle, viscosity solutions, robust optimal stopping, stopping strategies. 35 pages. Final version. To appear in the SIAM Journal on Control and Optimization