Electronic Journal of Probability

Optimal stopping time problem in a general framework

Magdalena Kobylanski and Marie-Claire Quenez

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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

Article information

Electron. J. Probab., Volume 17 (2012), paper no. 72, 28 pp.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]

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Kobylanski, Magdalena; Quenez, Marie-Claire. Optimal stopping time problem in a general framework. Electron. J. Probab. 17 (2012), paper no. 72, 28 pp. doi:10.1214/EJP.v17-2262. https://projecteuclid.org/euclid.ejp/1465062394

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