On the Optimal Stopping of a One-dimensional Diffusion

TL;DR

This paper analyzes the optimal stopping problem for a one-dimensional diffusion with Borel-measurable coefficients, providing new analytic representations and characterizations of the value function, including a generalized smooth fit principle.

Contribution

It introduces a novel analytic representation of the $r$-potential and characterizes the value function as a unique solution to a variational inequality for general reward functions.

Findings

New analytic representation of the $r$-potential.

Characterization of the value function via variational inequality.

Generalization of the principle of smooth fit.

Abstract

We consider a one-dimensional diffusion which solves a stochastic differential equation with Borel-measurable coefficients in an open interval. We allow for the endpoints to be inaccessible or absorbing. Given a Borel-measurable function $r$ that is uniformly bounded away from 0, we establish a new analytic representation of the $r$-potential of a continuous additive functional of the diffusion. We also characterize the value function of an optimal stopping problem with general reward function as the unique solution of a variational inequality (in the sense of distributions) with appropriate growth or boundary conditions. Furthermore, we establish several other characterisations of the solution to the optimal stopping problem, including a generalisation of the so-called "principle of smooth fit".