Dynkin games in a general framework

TL;DR

This paper revisits Dynkin games within a broad framework, improving classical results, relaxing assumptions, and establishing conditions for fairness, value functions, and saddle points using elementary probability theory.

Contribution

It introduces a general framework for Dynkin games, constructs supermartingale families, and relaxes assumptions for fairness and saddle point existence.

Findings

Dynkin games are fair under weak regularity assumptions.

The value function is characterized as the difference of supermartingale families.

Existence of saddle points is established under minimal additional conditions.

Abstract

We revisit the Dynkin game problem in a general framework, improve classical results and relax some assumptions. The criterion is expressed in terms of families of random variables indexed by stopping times. We construct two nonnegative supermartingales families $J$ and $J'$ whose finitness is equivalent to the Mokobodski's condition. Under some weak right-regularity assumption, the game is shown to be fair and $J-J'$ is shown to be the common value function. Existence of saddle points is derived under some weak additional assumptions. All the results are written in terms of random variables and are proven by using only classical results of probability theory.