On the monotonicity of the stopping boundary for time-inhomogeneous optimal stopping problems

TL;DR

This paper establishes conditions ensuring the monotonicity of the optimal stopping boundary in time-inhomogeneous problems driven by SDEs, using probabilistic and martingale methods, with implications for free-boundary problems.

Contribution

It provides new sufficient conditions for the monotonicity of the stopping boundary in time-inhomogeneous optimal stopping problems, extending previous results to more general settings.

Findings

Monotonicity of the stopping boundary is guaranteed under certain data conditions.

Comparison principles and martingale methods are effective tools in this analysis.

A variant of the main theorem connects optimal stopping with free-boundary problems.

Abstract

We consider a class of time-inhomogeneous optimal stopping problems and we provide sufficient conditions on the data of the problem that guarantee monotonicity of the optimal stopping boundary. In our setting, time-inhomogeneity stems not only from the reward function but, in particular, from the time dependence of the drift coefficient of the one-dimensional stochastic differential equation (SDE) which drives the stopping problem. In order to obtain our results, we mostly employ probabilistic arguments: we use a comparison principle between solutions of the SDE computed at different starting times, and martingale methods of optimal stopping theory. We also show a variant of the main theorem, which weakens one of the assumptions and additionally relies on the renowned connection between optimal stopping and free-boundary problems.