# A new integral equation for the first passage time density of the Ornstein-Uhlenbeck process

## TL;DR

This paper introduces a novel Volterra integral equation for the first passage time density of the Ornstein-Uhlenbeck process, utilizing inverse Laplace transforms of parabolic cylinder functions, applicable to constant and time-dependent thresholds.

## Contribution

It derives a new integral equation for the first passage time density using inverse Laplace transforms, extending the Fortet renewal equation and applicable to various thresholds.

## Key findings

- The integral equation is valid for constant and time-dependent thresholds.
- The kernel involves a parabolic cylinder function and is regular for q<=-1.
- The Fortet renewal equation is a special case of the new integral equation.

## Abstract

The Laplace transform of the first passage time density of the Ornstein--Uhlenbeck process for a constant threshold contains a ratio of two parabolic cylinder functions for which no analytical inversion formula is available. Recently derived inverse Laplace transforms for the product of two parabolic cylinder functions together with the convolution theorem of the Laplace transform then allow to derive a new Volterra integral equation for this first passage time density. The kernel of this integral equation contains a parabolic cylinder function and the Fortet renewal equation for the Ornstein-Uhlenbeck process emerges as a special case, namely when the order q of the parabolic cylinder function is set at 0. The integral equation is shown to hold both for constant as well as time dependent thresholds. Moreover, the kernel of the integral equation is regular for q<=-1.