A note on the continuity of free-boundaries in finite-horizon optimal stopping problems for one dimensional diffusions

TL;DR

This paper establishes sufficient conditions ensuring the continuity of free-boundaries in finite-horizon optimal stopping problems for one-dimensional diffusions, combining analytic and probabilistic methods.

Contribution

It provides new sufficient conditions for free-boundary continuity in finite-horizon stopping problems with one-dimensional diffusions, using PDE and stochastic analysis techniques.

Findings

Conditions for free-boundary continuity are identified.

The proof combines PDE maximum principles with probabilistic arguments.

Results apply to models in finance and economics.

Abstract

We provide sufficient conditions for the continuity of the free-boundary in a general class of finite-horizon optimal stopping problems arising for instance in finance and economics. The underlying process is a strong solution of one dimensional, time-homogeneous stochastic differential equation (SDE). The proof relies on both analytic and probabilistic arguments and it is based on a contradiction scheme inspired by the maximum principle in partial differential equations (PDE) theory. Mild, local regularity of the coefficients of the SDE and smoothness of the gain function locally at the boundary are required.