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

Miryana Grigorova (1) , Marie-Claire Quenez (2)

1 Institut für Mathematik [Humboldt]
2 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

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.

Mots clés: optimal stopping; non-zero-sum Dynkin game; g-expectation; dynamic risk measure; game option; Nash equilibrium

Domaines: Probabilités [math.PR]; Finance quantitative [q-fin.CP]; Gestion des risques [q-fin.RM]

Miryana Grigorova, Marie-Claire Quenez. Optimal stopping and a non-zero-sum Dynkin game in discrete time with risk measures induced by BSDEs. Stochastics: An International Journal of Probability and Stochastic Processes, 2017, 89 (1), ⟨10.1080/17442508.2016.1166505⟩. ⟨hal-01519215⟩