# Exact simulation of first exit times for one-dimensional diffusion processes

## TL;DR

This paper introduces an exact simulation algorithm for first exit times of one-dimensional diffusion processes, avoiding errors from discretization by using advanced stochastic techniques.

## Contribution

A novel acceptance-rejection algorithm that precisely simulates first exit times without approximation errors for one-dimensional diffusions.

## Key findings

- Algorithm accurately simulates first exit times
- The method is efficient and theoretically justified
- Numerical examples demonstrate practical effectiveness

## Abstract

The simulation of exit times for diffusion processes is a challenging task since it concerns many applications in different fields like mathematical finance, neuroscience, reliability... The usual procedure is to use discretiza-tion schemes which unfortunately introduce some error in the target distribution. Our aim is to present a new algorithm which simulates exactly the exit time for one-dimensional diffusions. This acceptance-rejection algorithm requires to simulate exactly the exit time of the Brownian motion on one side and the Brownian position at a given time, constrained not to have exit before, on the other side. Crucial tools in this study are the Girsanov transformation, the convergent series method for the simulation of random variables and the classical rejection sampling. The efficiency of the method is described through theoretical results and numerical examples.