Optimal multiple stopping time problem

Abstract

We study the optimal multiple stopping time problem defined for each stopping time S by ü(S) = ess sup?1,...,?d≥S E[ψ(?1, . . . , ?d )|F S ]. The key point is the construction of a new reward θ such that the üalue function ü(S) also satisfies ü(S) = ess supθ ≥S E[φ(θ)|F S ]. This new reward φ is not a right-continuous adapted process as in the classical case, but a family of random üariables. For such a reward, we proüe a new existence result for optimal stopping times under weaker assumptions than in the classical case. This result is used to proüe the existence of optimal multiple stopping times for ü(S) by a constructiüe method. Moreoüer, under strong regularity assumptions on ψ, we show that the new reward φ can be aggregated by a progressiüe process. This leads to new applications, particularly in finance (applications to American options with multiple exercise times). © 2011 Institute of Mathematical Statistics.