## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 27, Number 3 (2017), 1702-1755.

### On the robust Dynkin game

#### Abstract

We analyze a robust version of the Dynkin game over a set $\mathcal{P}$ of mutually singular probabilities. We first prove that conservative player’s lower and upper value coincide (let us denote the value by $V$). Such a result connects the robust Dynkin game with second-order doubly reflected backward stochastic differential equations. Also, we show that the value process $V$ is a submartingale under an appropriately defined nonlinear expectation $\underline{\mathscr{E}}$ up to the first time $\tau_{*}$ when $V$ meets the lower payoff process $L$. If the probability set $\mathcal{P}$ is weakly compact, one can even find an optimal triplet $(\mathbb{P}_{*},\tau_{*},\gamma_{*})$ for the value $V_{0}$.

The mutual singularity of probabilities in $\mathcal{P}$ causes major technical difficulties. To deal with them, we use some new methods including two approximations with respect to the set of stopping times.

#### Article information

**Source**

Ann. Appl. Probab. Volume 27, Number 3 (2017), 1702-1755.

**Dates**

Received: July 2015

Revised: April 2016

First available in Project Euclid: 19 July 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1500451239

**Digital Object Identifier**

doi:10.1214/16-AAP1243

**Subjects**

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 93E20: Optimal stochastic control 49L20: Dynamic programming method 91A15: Stochastic games 91A55: Games of timing 60G44: Martingales with continuous parameter 91B28

**Keywords**

Robust Dynkin game nonlinear expectation dynamic programming principle controls in weak formulation weak stability under pasting martingale approach path-dependent stochastic differential equations with controls optimal triplet optimal stopping with random maturity

#### Citation

Bayraktar, Erhan; Yao, Song. On the robust Dynkin game. Ann. Appl. Probab. 27 (2017), no. 3, 1702--1755. doi:10.1214/16-AAP1243. https://projecteuclid.org/euclid.aoap/1500451239.