The Annals of Applied Probability

On the robust Dynkin game

Erhan Bayraktar and Song Yao

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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.


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