From optimal martingales to randomized dual optimal stopping

TL;DR

This paper classifies optimal martingales in dual optimal stopping problems, highlighting the robustness of the Doob martingale and introducing a randomized algorithm for efficient dual bounds without nested simulation.

Contribution

It introduces a novel randomized dual martingale minimization algorithm that efficiently approximates the Doob martingale, improving dual bounds in optimal stopping problems.

Findings

The Doob martingale is the most robust surely optimal martingale.

The new algorithm avoids nested simulation and efficiently finds near-Doob martingales.

The approach yields low-variance dual upper bounds for optimal stopping problems.

Abstract

In this article we study and classify optimal martingales in the dual formulation of optimal stopping problems. In this respect we distinguish between weakly optimal and surely optimal martingales. It is shown that the family of weakly optimal and surely optimal martingales may be quite large. On the other hand it is shown that the Doob-martingale, that is, the martingale part of the Snell envelope, is in a certain sense the most robust surely optimal martingale under random perturbations. This new insight leads to a novel randomized dual martingale minimization algorithm that doesn't require nested simulation. As a main feature, in a possibly large family of optimal martingales the algorithm efficiently selects a martingale that is as close as possible to the Doob martingale. As a result, one obtains the dual upper bound for the optimal stopping problem with low variance.