Exact simulation of Brownian diffusions with drift admitting jumps

TL;DR

This paper introduces an exact simulation method for Brownian diffusions with jumpy drifts using retrospective rejection sampling, explicitly handling the complex case of two jumps with a new transition density expression.

Contribution

It provides a novel explicit transition density for skew Brownian motion with two barriers, enabling exact simulation of diffusions with jumps in the drift.

Findings

Successful numerical simulations demonstrating the method.

Explicit transition density formula for skew Brownian motion with two barriers.

Enhanced ability to simulate diffusions with complex jump structures.

Abstract

In this paper, using an algorithm based on the retrospective rejection sampling scheme, we propose an exact simulation of a Brownian diffusion whose drift admits several jumps. We treat explicitly and extensively the case of two jumps, providing numerical simulations. Our main contribution is to manage the technical diffculty due to the presence of two jumps thanks to a new explicit expression of the transition density of the skew Brownian motion with two semipermeable barriers and a constant drift.