On the first hitting times of one dimensional elliptic diffusions

TL;DR

This paper studies the distribution of the first hitting times of thresholds by one-dimensional elliptic diffusions, providing explicit formulas, regularity results, and Gaussian estimates useful for simulations.

Contribution

It introduces a parametrix-based method to explicitly characterize the law of first hitting times and the stopped process, with regularity and Gaussian bounds under mild conditions.

Findings

Explicit transition density expressions for hitting times

Regularity properties up to the boundary

Gaussian upper estimates for the law

Abstract

In this article, we obtain properties of the law associated to the first hitting time of a threshold by a one-dimensional uniformly elliptic diffusion process and to the associated process stopped at the threshold. Our methodology relies on the parametrix method that we apply to the associated Markov semigroup. It allows to obtain explicit expressions for the corresponding transition densities and to study its regularity properties up to the boundary under mild assumptions on the coefficients. As a by product, we also provide Gaussian upper estimates for these laws and derive a probabilistic representation that may be useful for the construction of an unbiased Monte Carlo path simulation method, among other applications.