Optimal multiple stopping time problem

Details Optimal multiple stopping time problem Optimal multiple stopping time problem

2009-10-15

arxiv.org/abs/0910.2788v3

Annals of Applied Probability 2011, Vol. 21, No. 4, 1365-1399

Mathematics

Probability

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

Scientific paper

10.1214/10-AAP727

We study the optimal multiple stopping time problem defined for each stopping time $S$ by $v(S)=\operatorname {ess}\sup_{\tau_1,...,\tau_d\geq S}E[\psi(\tau_1,...,\tau_d)|\mathcal{F}_S]$. The key point is the construction of a new reward $\phi$ such that the value function $v(S)$ also satisfies $v(S)=\operatorname {ess}\sup_{\theta\geq S}E[\phi(\theta)|\mathcal{F}_S]$. 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 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 $\phi$ can be aggregated by a progressive process. This leads to new applications, particularly in finance (applications to American options with multiple exercise times).

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