Optimal multiple stopping time problem

Abstract

We study the optimal multiple stopping time problem defined for each stopping time $S$ by $v(S)=\operatorname {ess}\sup_{τ_1,...,τ_d\geq S}E[ψ(τ_1,...,τ_d)|\mathcal{F}_S]$. The key point is the construction of a new reward $ϕ$ such that the value function $v(S)$ also satisfies $v(S)=\operatorname {ess}\sup_{θ\geq S}E[ϕ(θ)|\mathcal{F}_S]$. 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).