Martingale theory for Dynkin games with asymmetric information
Tiziano De Angelis, Jan Palczewski, Jacob Smith
TL;DR
This work develops a general, non-Markovian framework for zero-sum Dynkin games with asymmetric information, where two players observe different filtrations. It introduces a dynamic, martingale-based characterization of equilibrium payoffs via optional semimartingales and Doob–Meyer decompositions, and provides necessary and sufficient saddle-point conditions expressed through auxiliary martingale systems and generating processes. The authors illuminate the structure of optimal randomised stopping times and offer two natural application classes—partially observed regimes and dynamics—with computable expressions and belief-driven strategies. A heuristic PDE link helps connect the martingale method to classical variational inequalities, enriching both the theory and potential numerical approaches for complex, information-asymmetric stopping games.
Abstract
This paper provides necessary and sufficient conditions for a pair of randomised stopping times to form a saddle point of a zero-sum Dynkin game with partial and/or asymmetric information across players. The framework is non-Markovian and covers essentially any information structure. Our methodology relies on the identification of suitable super and submartingales involving players' equilibrium payoffs. Saddle point strategies are characterised in terms of the dynamics of those equilibrium payoffs and are related to their Doob-Meyer decompositions.