Optimal stopping of an Ornstein-Uhlenbeck bridge

TL;DR

This paper rigorously analyzes the optimal stopping problem for an Ornstein-Uhlenbeck bridge, characterizing the free boundary and providing numerical methods, with implications for related Brownian bridge problems.

Contribution

It introduces a novel time-space transformation approach to analyze the optimal stopping boundary for Ornstein-Uhlenbeck bridges, including the Brownian bridge as a special case.

Findings

The free boundary shape varies and is generally non-monotonic.

Numerical algorithms effectively compute the free boundary.

The methodology simplifies the problem to an infinite horizon with Brownian motion.

Abstract

We make a rigorous analysis of the existence and characterization of the free boundary related to the optimal stopping problem that maximizes the mean of an Ornstein--Uhlenbeck bridge. The result includes the Brownian bridge problem as a limit case. The methodology hereby presented relies on a time-space transformation that casts the original problem into a more tractable one with an infinite horizon and a Brownian motion underneath. We comment on two different numerical algorithms to compute the free-boundary equation and discuss illustrative cases that shed light on the boundary's shape. In particular, the free boundary generally does not share the monotonicity of the Brownian bridge case.