# Optimally Stopping a Brownian Bridge with an Unknown Pinning Time: A Bayesian Approach

## TL;DR

This paper develops a Bayesian framework for optimally stopping a Brownian bridge with an unknown pinning time, providing explicit solutions for gamma and beta priors and revealing time-homogeneity and boundary structures.

## Contribution

It introduces a Bayesian approach to the stopping problem with unknown pinning time, deriving closed-form solutions and characterizing the stopping boundary for specific priors.

## Key findings

- Gamma prior leads to a time-homogeneous, fully solvable problem.
- Beta prior results in a square-root boundary similar to known pinning time case.
- Explicit solutions enhance understanding of optimal stopping under uncertainty.

## Abstract

We consider the problem of optimally stopping a Brownian bridge with an unknown pinning time so as to maximise the value of the process upon stopping. Adopting a Bayesian approach, we assume the stopper has a general continuous prior and is allowed to update their belief about the value of the pinning time through sequential observations of the process. Uncertainty in the pinning time influences both the conditional dynamics of the process and the expected (random) horizon of the optimal stopping problem. We analyse certain gamma and beta distributed priors in detail. Remarkably, the optimal stopping problem in the gamma case becomes time homogeneous and is completely solvable in closed form. Moreover, in the beta case we find that the optimal stopping boundary takes on a square-root form, similar to the classical solution with a known pinning time.