Continuous-time Zero-Sum Stochastic Game with Stopping and Control

TL;DR

This paper studies a continuous-time zero-sum stochastic game involving Markov chains with unbounded rates, where players can also choose to stop the process, establishing the existence of a value and saddle point equilibrium.

Contribution

It introduces a framework for zero-sum stochastic games with stopping in continuous time and proves the existence of a unique value and equilibrium under certain conditions.

Findings

The game has a well-defined value.

Existence of a saddle point equilibrium.

Solution characterized by dynamic programming inequalities.

Abstract

We consider a zero-sum stochastic game for continuous-time Markov chain with countable state space and unbounded transition and pay-off rates. The additional feature of the game is that the controllers together with taking actions are also allowed to stop the process. Under suitable hypothesis we show that the game has a value and it is the unique solution of certain dynamic programming inequalities with bilateral constraints. In the process we also prescribe a saddle point equilibrium.