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2012 Optimal stopping time problem in a general framework
Magdalena Kobylanski, Marie-Claire Quenez
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Electron. J. Probab. 17: 1-28 (2012). DOI: 10.1214/EJP.v17-2262


We study the optimal stopping time problem $v(S)={\rm ess}\sup_{\theta \geq S} E[\phi(\theta)|\mathcal{F}_S]$, for any stopping time $S$, where the reward is given by a family $(\phi(\theta),\theta\in\mathcal{T}_0)$ of non negative random variables indexed by stopping times. We solve the problem under weak assumptions in terms of integrability and regularity of the reward family. More precisely, we only suppose $v(0) < + \infty$ and $(\phi(\theta),\theta\in \mathcal{T}_0)$ upper semicontinuous along stopping times in expectation. We show the existence of an optimal stopping time and obtain a characterization of the minimal and the maximal optimal stopping times. We also provide some local properties of the value function family. All the results are written in terms of families of random variables and are proven by only using classical results of the Probability Theory


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Magdalena Kobylanski. Marie-Claire Quenez. "Optimal stopping time problem in a general framework." Electron. J. Probab. 17 1 - 28, 2012.


Accepted: 29 August 2012; Published: 2012
First available in Project Euclid: 4 June 2016

zbMATH: 06098155
MathSciNet: MR2968679
Digital Object Identifier: 10.1214/EJP.v17-2262

Primary: 60G40

Keywords: American options , Optimal stopping , supermartingale


Vol.17 • 2012
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