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On a stopping game in continuous time. (English) Zbl 1345.60037

The paper deals with a stopping game where both players’ stopping decisions have impact on the payoff. On a filtered probability space \((\Omega,\mathcal{F},P,\mathbb{F}=(\mathcal{F}_t)_{t=0,\ldots,T})\), a gain function \(U(s,t)\) is defined, which is \(\mathcal{F}_{s\vee t}\)-measurable. Let \(\mathcal{T}\) be the set of stopping times. \(\rho\) and \(\tau\) are mappings from \(\mathcal T\;\) to \(\mathcal{T}\) satisfying certain non-anticipativity conditions. Define \(\overline C:=\inf_{\rho}\sup_{\tau\in\mathcal T}\mathbf{E}[U(\rho(\tau),\tau)]\) and \(\underline C:=\sup_{\tau}\inf_{\rho\in\mathcal T}\mathbf{E}[U(\rho,\tau(\rho))]\). By converting the problems into a corresponding Dynkin game it is shown that \(\overline C=\underline C\). The results can be applied to weaken the assumption on path regularities on the reward processes in a paper by M. Kobylanski et al. [Stochastics 86, No. 2, 304–329 (2014; Zbl 1298.60050); corrigendum ibid. 86, No. 2, 370 (2014; Zbl 1298.60051)].

MSC:

60G40 Stopping times; optimal stopping problems; gambling theory
91A60 Probabilistic games; gambling
91A10 Noncooperative games
60G07 General theory of stochastic processes
93E20 Optimal stochastic control
37A50 Dynamical systems and their relations with probability theory and stochastic processes
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References:

[1] Zhou1 Erhan Bayraktar and Zhou Zhou, On zero-sum optimal stopping games, (2014), Preprint, arXiv:1408.3692. · Zbl 1404.49027
[2] Karatzas, Ioannis; Shreve, Steven E., Methods of mathematical finance, Applications of Mathematics (New York) 39, xvi+407 pp. (1998), Springer-Verlag, New York · Zbl 0941.91032 · doi:10.1007/b98840
[3] Kobylanski, Magdalena; Quenez, Marie-Claire, Optimal stopping time problem in a general framework, Electron. J. Probab., 17, no. 72, 28 pp. (2012) · Zbl 1405.60055 · doi:10.1214/EJP.v17-2262
[4] Kobylanski, Magdalena; Quenez, Marie-Claire; de Campagnolle, Marc Roger, Dynkin games in a general framework, Stochastics, 86, 2, 304-329 (2014) · Zbl 1298.60050 · doi:10.1080/17442508.2013.778860
[5] Kobylanski, Magdalena; Quenez, Marie-Claire; Rouy-Mironescu, Elisabeth, Optimal multiple stopping time problem, Ann. Appl. Probab., 21, 4, 1365-1399 (2011) · Zbl 1235.60040 · doi:10.1214/10-AAP727
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