# Optimal multiple stopping time problem

Abstract : We study the optimal multiple stopping time problem defined for each stopping time $S$ by $\displaystyle{ v(S)=\esssup_ {\tau_1,\cdots,\tau_d \geq S } E[\psi( \tau_1,\cdots,\tau_d)\, |\,\F_S]\,}$.\\ The key point is the construction of a {\em new reward} $\phi$ such that the value function $v(S)$ satisfies also $v(S)=\esssup_ {\theta \geq S } \,E[\phi(\theta)\, |\,\F_S]\,.$ This new reward $\phi$ is not a right continuous adapted process as in the classical case but a family of random variables. For such a reward, we prove a new existence result of optimal stopping times under weaker assumptions than in the classical case. This result is used to prove the existence of optimal multiple stopping times for $v(S)$ by a constructive method. Moreover, under strong regularity assumptions on $\psi$, we show that the new reward $\phi$ can be aggregated by a progressive process. This leads to different applications in particular in finance for American options with multiple exercise times.
Keywords :
Type de document :
Pré-publication, Document de travail
2009
Domaine :

Littérature citée [13 références]

https://hal.archives-ouvertes.fr/hal-00423896
Contributeur : Magdalena Kobylanski <>
Soumis le : mercredi 9 juin 2010 - 15:18:14
Dernière modification le : mercredi 21 mars 2018 - 18:56:49
Document(s) archivé(s) le : jeudi 23 septembre 2010 - 12:43:14

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• HAL Id : hal-00423896, version 2
• ARXIV : 0910.2788

### Citation

Magdalena Kobylanski, Marie-Claire Quenez, Elisabeth Rouy-Mironescu. Optimal multiple stopping time problem. 2009. 〈hal-00423896v2〉

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