On the controller-stopper problems with controlled jumps

TL;DR

This paper extends the analysis of controller-stopper games to include controlled jumps in the state process, using the Stochastic Perron method to address the challenges posed by game-theoretic dynamic programming.

Contribution

It introduces a novel approach to analyze controller-stopper games with controlled jumps, bypassing the need for dynamic programming principles.

Findings

Resolved stochastic target problems with controlled jumps.

Embedded original problems into stochastic target frameworks.

Demonstrated the effectiveness of the Stochastic Perron method for games.

Abstract

We analyze the continuous time zero-sum and cooperative controller-stopper games of Karatzas and Sudderth [Annals of Probability, 2001], Karatzas and Zamfirescu [Annals of Probability, 2008] and Karatzas and Zamfirescu [Applied Mathematics and Optimization, 2005] when the volatility of the state process is controlled as in Bayraktar and Huang [SIAM Journal on Control and Optimization, 2013] but additionally when the state process has controlled jumps. We perform this analysis by first resolving the stochastic target problems (of Soner and Touzi [SIAM Journal on Control and Optimization, 2002; Journal of European Mathematical Society, 2002]) with a cooperative or a non-cooperative stopper and then embedding the original problem into the latter set-up. Unlike in Bayraktar and Huang [SIAM Journal on Control and Optimization, 2013] our analysis relies crucially on the Stochastic Perron method of Bayraktar and Sirbu [SIAM Journal on Control and Optimization, 2013] but not the dynamic programming principle, which is difficult to prove directly for games.