Threshold Diffusions

TL;DR

This paper introduces threshold diffusion processes defined by stochastic differential equations with step-function coefficients, providing explicit formulas for hitting times and potential measures, and analyzing their long-term behaviors.

Contribution

It presents the first explicit solutions for threshold diffusions' hitting times and potential measures, advancing understanding of their asymptotic properties.

Findings

Explicit formulas for hitting times and potential measures.

Analysis of stationary distributions.

Discussion of escape probabilities.

Abstract

We propose threshold diffusion processes as unique solutions to stochastic differential equations with step-function coefficients, and obtain explicit expressions for the conditional Laplace transform of the hitting times and the potential measures. Applying these results, we further discuss their asymptotic behaviors such as the stationary distributions and the escape probabilities.