Optimal double stopping time

Abstract

We consider the optimal double stopping time problem defined for each stopping time $S$ by $v(S)=\esssup\{E[ψ(τ_1, τ_2) | \F_S], τ_1, τ_2 \geq S \}$. Following the optimal one stopping time problem, we study the existence of optimal stopping times and give a method to compute them. The key point is the construction of a {\em new reward} $ϕ$ such that the value function $v(S)$ satisfies $v(S)=\esssup\{E[ϕ(τ) | \F_S], τ\geq S \}$. Finally, we give an example of an american option with double exercise time.