Optimal stopping in a general framework

Magdalena Kobylanski (1) , Marie-Claire Quenez (2)

1 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
2 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

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 $(\phi(\theta),\theta\in\mathcal{T}_0)$ \emph{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.

Mots clés: Optimal stopping; supermartingale; american options

Domaines: Probabilités [math.PR]

Magdalena Kobylanski, Marie-Claire Quenez. Optimal stopping in a general framework. Electronic Journal of Probability, 2012, 17 (72), pp.1-28. ⟨hal-00519457v3⟩