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

TL;DR

This paper establishes the existence of a value and optimal strategies in non-Markovian, continuous-time zero-sum Dynkin games with partial and asymmetric information, using a probabilistic approach rather than PDE methods.

Contribution

It introduces a novel probabilistic and functional analytic framework for Dynkin games with general information structures, extending beyond Markovian assumptions.

Findings

Proves the existence of a game value in non-Markovian settings.

Establishes the existence of optimal strategies for both players.

Provides counterexamples illustrating the limits of the assumptions.

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 \cadlag 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.