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June 2017 On the robust Dynkin game
Erhan Bayraktar, Song Yao
Ann. Appl. Probab. 27(3): 1702-1755 (June 2017). DOI: 10.1214/16-AAP1243


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.


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Erhan Bayraktar. Song Yao. "On the robust Dynkin game." Ann. Appl. Probab. 27 (3) 1702 - 1755, June 2017.


Received: 1 July 2015; Revised: 1 April 2016; Published: June 2017
First available in Project Euclid: 19 July 2017

zbMATH: 1371.60071
MathSciNet: MR3678483
Digital Object Identifier: 10.1214/16-AAP1243

Primary: 49L20 , 60G40 , 60G44 , 91A15 , 91A55 , 91B28 , 93E20

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

Rights: Copyright © 2017 Institute of Mathematical Statistics


Vol.27 • No. 3 • June 2017
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