On the optimal stopping of Gauss-Markov bridges with random pinning points

TL;DR

This paper studies the optimal stopping problem for Gauss-Markov processes with prescribed terminal distributions, transforming it into a Brownian bridge problem and analyzing the structure of optimal stopping boundaries.

Contribution

It introduces a time-space transformation to analyze the problem, establishes Lipschitz continuity of the value function, and provides conditions for the boundary to be a function.

Findings

Optimal stopping rule is the first entry into the stopping region.

Value function is Lipschitz continuous on compact sets.

Boundaries can be bounded and structured under certain density conditions.

Abstract

We consider the optimal stopping problem for a Gauss-Markov process conditioned to adopt a prescribed terminal distribution. By applying a time-space transformation, we show it is equivalent to stopping a Brownian bridge pinned at a random endpoint with a time-dependent payoff. We prove that the optimal rule is the first entry into the stopping region, and establish that the value function is Lipschitz continuous on compacts via a coupling of terminal pinning points across different initial conditions. A comparison theorems then order value functions according to likelihood-ratio ordering of terminal densities, and when these densities have bounded support, we bound the optimal boundary by that of a Gauss-Markov bridge. Although the stopping boundary need not be the graph of a function in general, we provide sufficient conditions under which this property holds, and identify strongly log-concave terminal densities that guarantee this structure. Numerical experiments illustrate representative boundary shapes.