From optimal stopping boundaries to Rost's reversed barriers and the Skorokhod embedding

TL;DR

This paper offers a new probabilistic proof linking Rost's solution to the Skorokhod embedding problem with optimal stopping problems for Brownian motion, demonstrating that time reversal of stopping sets forms Rost's reversed barrier.

Contribution

It introduces a novel probabilistic approach using stochastic calculus to connect Rost's solution with optimal stopping problems and reversed barriers.

Findings

Proves the connection between Rost's solution and optimal stopping sets.

Shows that the time reversal of stopping sets forms Rost's reversed barrier.

Provides a new probabilistic proof using stochastic calculus.

Abstract

We provide a new probabilistic proof of the connection between Rost's solution of the Skorokhod embedding problem and a suitable family of optimal stopping problems for Brownian motion with finite time-horizon. In particular we use stochastic calculus to show that the time reversal of the optimal stopping sets for such problems forms the so-called Rost's reversed barrier.