# Exact simulation of the first-passage time of diffusions

## TL;DR

This paper introduces an exact rejection sampling algorithm for simulating the first-passage time of general one-dimensional diffusion processes efficiently, avoiding full path simulation and leveraging Girsanov's transformation.

## Contribution

The authors develop a novel rejection sampling method for exact simulation of first-passage times in diffusions, extending beyond Brownian cases.

## Key findings

- Algorithm achieves exact simulation without full path construction.
- Efficiency is supported by theoretical analysis and numerical tests.
- Method applies to a wide class of one-dimensional diffusions.

## Abstract

Since diffusion processes arise in so many different fields, efficient tech-nics for the simulation of sample paths, like discretization schemes, represent crucial tools in applied probability. Such methods permit to obtain approximations of the first-passage times as a by-product. For efficiency reasons, it is particularly challenging to simulate directly this hitting time by avoiding to construct the whole paths. In the Brownian case, the distribution of the first-passage time is explicitly known and can be easily used for simulation purposes. The authors introduce a new rejection sampling algorithm which permits to perform an exact simulation of the first-passage time for general one-dimensional diffusion processes. The efficiency of the method, which is essentially based on Girsanov's transformation , is described through theoretical results and numerical examples.