Revisiting a Theorem of L.A. Shepp on Optimal Stopping

TL;DR

This paper revisits a classical optimal stopping theorem using a bond-selling example, providing a new proof based on martingale methods that emphasizes the importance of martingale theory in such proofs.

Contribution

It offers a novel proof of Shepp's theorem on optimal stopping, highlighting the role of martingales in establishing optimality.

Findings

New proof of Shepp's theorem using martingale techniques

Demonstrates the effectiveness of guessing and verifying control functions

Reinforces the significance of martingale theory in optimal stopping problems

Abstract

Using a bondholder who seeks to determine when to sell his bond as our motivating example, we revisit one of Larry Shepp's classical theorems on optimal stopping. We offer a novel proof of Theorem 1 from from \cite{Shepp}. Our approach is that of guessing the optimal control function and proving its optimality with martingales. Without martingale theory one could hardly prove our guess to be correct.