Optimal multiple stopping time problem

Abstract

We study the optimal multiple stopping time problem defined for each stopping time S by $v(S)=\operatorname{ess}\sup_{\tau_{1},\ldots,\tau_{d}\geq S}E[\psi(\tau_{1},\ldots,\tau_{d})|\mathcal{F}_{S}]$.

The key point is the construction of a new reward ϕ 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 ϕ 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 ψ, we show that the new reward ϕ can be aggregated by a progressive process. This leads to new applications, particularly in finance (applications to American options with multiple exercise times).