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

TL;DR

This paper investigates optimal stopping and non-zero-sum Dynkin games in discrete time where risk is evaluated through g-expectations derived from BSDEs, providing a recursive method to find Nash equilibria.

Contribution

It introduces a novel approach to solve non-zero-sum Dynkin games with risk measures from BSDEs, constructing Nash equilibria without extra assumptions on the driver.

Findings

Established existence of Nash equilibrium in the game.

Developed a recursive procedure for equilibrium construction.

Extended results to general Lipschitz drivers.

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. Hamad{\`e}ne 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.