On the value of non-Markovian Dynkin games with partial and asymmetric information

Abstract

We prove that zero-sum Dynkin games in continuous time with partial and asymmetric information admit a value in randomised stopping times when the stopping payoffs of the players are general càdlàg measurable processes. As a by-product of our method of proof we also obtain existence of optimal strategies for both players. The main novelties are that we do not assume a Markovian nature of the game nor a particular structure of the information available to the players. This allows us to go beyond the variational methods (based on PDEs) developed in the literature on Dynkin games in continuous time with partial/asymmetric information. Instead, we focus on a probabilistic and functional analytic approach based on the general theory of stochastic processes and Sion's min-max theorem (M. Sion, Pacific J. Math., 8, 1958, pp. 171-176). Our framework encompasses examples found in the literature on continuous time Dynkin games with asymmetric information and we provide counterexamples to show that our assumptions cannot be further relaxed.

Numerical Approximation of the Value of a Stochastic Differential Game with Asymmetric Information

“…This in particular implies asymmetric information, in the sense that players employ stopping rules that are stopping times with respect to different filtrations. More recently, in a very general not necessarily Markovian setting, it is proved in [9] that continuous-time zero-sum Dynkin games with partial and asymmetric information admit a value in randomized stopping times. As a byproduct, existence of equilibrium strategies for both players are also shown to exist.…”