# Optimal stopping of a Brownian bridge with an unknown pinning point

## TL;DR

This paper investigates the optimal stopping problem for a Brownian bridge with an unknown pinning point, analyzing how prior information influences the stopping strategy and characterizing the structure of the optimal stopping region.

## Contribution

It provides new insights into the structure of the optimal stopping region depending on different priors, including two-point and mixed Gaussian cases.

## Key findings

- Establishes continuity and bounds of the value function.
- Identifies conditions for one-sided stopping regions.
- Reveals complex structures in specific prior cases.

## Abstract

The problem of stopping a Brownian bridge with an unknown pinning point to maximise the expected value at the stopping time is studied. A few general properties, such as continuity and various bounds of the value function, are established. However, structural properties of the optimal stopping region are shown to crucially depend on the prior, and we provide a general condition for a one-sided stopping region. Moreover, a detailed analysis is conducted in the cases of the two-point and the mixed Gaussian priors, revealing a rich structure present in the problem.