# Optimal stopping for the exponential of a Brownian bridge

## TL;DR

This paper addresses an optimal stopping problem for a Brownian bridge to maximize the expected exponential gain, introducing new techniques to handle the non-linear structure and deriving the optimal stopping rule.

## Contribution

It develops novel methods combining pathwise properties and martingale techniques to solve a non-linear optimal stopping problem for the Brownian bridge.

## Key findings

- Derived the optimal stopping rule for the exponential of a Brownian bridge.
- Proved regularity properties of the value function.
- Extended the methodology to handle non-linear gain functions.

## Abstract

In this paper we study the problem of stopping a Brownian bridge $X$ in order to maximise the expected value of an exponential gain function. In particular, we solve the stopping problem $$\sup_{0\le \tau\le 1}\mathsf{E}[\mathrm{e}^{X_\tau}]$$ which was posed by Ernst and Shepp in their paper [Commun. Stoch. Anal., 9 (3), 2015, pp. 419--423] and was motivated by bond selling with non-negative prices. Due to the non-linear structure of the exponential gain, we cannot rely on methods used in the literature to find closed-form solutions to other problems involving the Brownian bridge. Instead, we develop techniques that use pathwise properties of the Brownian bridge and martingale methods of optimal stopping theory in order to find the optimal stopping rule and to show regularity of the value function.