Optimal multiple stopping time problem

TL;DR

This paper introduces a novel approach to the optimal multiple stopping time problem by constructing a new reward family, enabling the proof of existence of optimal stopping times under weaker conditions, with applications in finance.

Contribution

It develops a new reward construction for multiple stopping problems, proving existence of optimal strategies under weaker assumptions and applying it to American options with multiple exercises.

Findings

New existence result for optimal stopping times

Construction of a reward family not necessarily right-continuous

Application to American options with multiple exercise times

Abstract

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).