Optimal stopping and a non-zero-sum Dynkin game in discrete time with risk measures induced by BSDEs

Abstract

We first study an optimal stopping problem in which a player (an agent) uses a discrete stopping time in order to stop optimally a payoff process whose risk is evaluated by a (non-linear) g-expectation. We then consider a non-zero-sum game on discrete stopping times with two agents who aim at minimizing their respective risks. The payoffs of the agents are assessed by g-expectations (with possibly different drivers for the different players). By using the results of the first part, combined with some ideas of S. Hamadène and J. Zhang, we construct a Nash equilibrium point of this game by a recursive procedure. Our results are obtained in the case of a standard Lipschitz driver g without any additional assumption on the driver besides that ensuring the monotonicity of the corresponding g-expectation.

Keywords:

• Optimal stopping
• non-zero-sum Dynkin game
• g-expectation
• dynamic risk measure
• game option
• Nash equilibrium

Notes

No potential conflict of interest was reported by the authors.