Martingale approach to stochastic differential games of control and stopping

TL;DR

This paper introduces a martingale-based method for analyzing non-Markovian stochastic differential games involving control and stopping, establishing the existence of a game value and optimal strategies.

Contribution

It develops a novel martingale approach for non-Markovian control and stopping games, characterizing saddle points through pathwise and martingale properties.

Findings

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

Existence of saddle pairs of optimal strategies is proven.

Characterization of saddle pairs via martingale properties.

Abstract

We develop a martingale approach for studying continuous-time stochastic differential games of control and stopping, in a non-Markovian framework and with the control affecting only the drift term of the state-process. Under appropriate conditions, we show that the game has a value and construct a saddle pair of optimal control and stopping strategies. Crucial in this construction is a characterization of saddle pairs in terms of pathwise and martingale properties of suitable quantities.