Explicit Asymptotics on First Passage Times of Diffusion Processes

TL;DR

This paper develops a unified framework using potential and perturbation theories to derive explicit asymptotic formulas for the first passage times of diffusion processes, with applications to several well-known stochastic models.

Contribution

It introduces a novel, unified approach for obtaining closed-form solutions for first passage times of diffusion processes, applicable to a broad class of models.

Findings

Closed-form approximations for Ornstein-Uhlenbeck and Bessel processes.

Application to exponential-Shiryaev process.

Numerical validation of the theoretical results.

Abstract

We introduce a unified framework for solving first passage times of time-homogeneous diffusion processes. According to the killed version potential theory and the perturbation theory, we are able to deduce closed-form solutions for probability densities of single-sided level crossing problem. The framework is applicable to diffusion processes with continuous drift functions, and a recursive system in the frequency domain has been provided. Besides, we derive a probabilistic representation for error estimation. The representation can be used to evaluate deviations in perturbed density functions. In the present paper, we apply the framework to Ornstein-Uhlenbeck and Bessel processes to find closed-form approximations for their first passage times; another successful application is given by the exponential-Shiryaev process. Numerical results are provided at the end of this paper.