Mokobodzki's intervals: an approach to Dynkin games when value process is not a semimartingale

TL;DR

This paper develops a new approach to Dynkin games with non-semimartingale payoff processes by introducing Mokobodzki's stochastic intervals and generalized reflected BSDEs, establishing existence, uniqueness, and game value properties.

Contribution

It introduces Mokobodzki's stochastic intervals and extends reflected BSDE theory to handle Dynkin games without the classical Mokobodzki's condition, providing new existence and uniqueness results.

Findings

Existence and uniqueness of solutions for generalized RBSDEs.

Establishment of the value process and saddle points for the game.

Convergence of the penalty scheme for the Dynkin game.

Abstract

We study Dynkin games governed by a nonlinear $\mathbb E^f$-expectation on a finite interval $[0,T]$, with payoff c\`adl\`ag processes $L,U$ of class (D) which are not imposed to satisfy (weak) Mokobodzki's condition - the existence of a c\`adl\`ag semimartingale between the barriers. For that purpose we introduce the notion of Mokobodzki's stochastic intervals $\mathscr M(\theta)$ (roughly speaking, maximal stochastic interval on which Mokobodzki's condition is satisfied when starting from the stopping time $\theta$) and the notion of reflected BSDEs without Mokobodzki's condition (this is a generalization and modification of the notion introduced by Hamad\'ene and Hassani (2005)). We prove an existence and uniqueness result for RBSDEs with driver $f$ that is non-increasing with respect to the value variable (no restrictions on the growth) and Lipschitz continuous with respect to the control variable, and with data in $L^1$ spaces. Next, by using RBSDEs, we show numerous results on Dynkin games: existence of the value process, saddle points, and convergence of the penalty scheme. We also show that the game is not played beyond $\mathscr M(\theta)$, when starting from $\theta$.