Optimal double stopping time

TL;DR

This paper investigates the problem of determining optimal double stopping times in stochastic processes, providing a method to compute them and illustrating with an American option example with double exercise rights.

Contribution

It introduces a new reward construction that reduces the double stopping problem to a single stopping problem, facilitating computation.

Findings

Existence of optimal double stopping times established.

A method to compute optimal stopping times via a new reward function.

Application to American options with double exercise rights.

Abstract

We consider the optimal double stopping time problem defined for each stopping time $S$ by $v(S)=\esssup\{E[\psi(\tau_1, \tau_2) | \F_S], \tau_1, \tau_2 \geq S \}$. 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} $\phi$ such that the value function $v(S)$ satisfies $v(S)=\esssup\{E[\phi(\tau) | \F_S], \tau \geq S \}$. Finally, we give an example of an american option with double exercise time.