Optimal multiple stopping time problem

Abstract

We study the optimal multiple stopping time problem defined for each stopping time $S$ by $v(S)=\mathrm{ess}{\sup }_{{\tau }_1,...,{\tau }_d\ge S}E[\psi ({\tau }_1,...,{\tau }_d)|{\mathcal{F}}_S]$. The key point is the construction of a new reward $\varphi$ such that the value function $v(S)$ also satisfies $v(S)=\mathrm{ess}{\sup }_{\theta \ge S}E[\varphi (\theta )|{\mathcal{F}}_S]$. This new reward $\varphi$ 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 for 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 $\varphi$ can be aggregated by a progressive process. This leads to new applications, particularly in finance (applications to American options with multiple exercise times).

Published in at http://dx.doi.org/10.1214/10-AAP727 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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