On the exact and $\varepsilon$-strong simulation of (jump) diffusions

TL;DR

This paper presents a framework for exact simulation of (jump) diffusion paths over finite intervals, enabling precise path reconstruction and barrier crossing analysis without discretisation errors.

Contribution

It extends existing exact algorithms with novel adaptive methods for simulating constrained (jump) diffusion paths within specified tolerances.

Findings

Able to simulate diffusion paths with arbitrary accuracy

Can determine barrier crossing events precisely

Efficiently simulate first hitting times and killed processes

Abstract

This paper introduces a framework for simulating finite dimensional representations of (jump) diffusion sample paths over finite intervals, without discretisation error (exactly), in such a way that the sample path can be restored at any finite collection of time points. Within this framework we extend existing exact algorithms and introduce novel adaptive approaches. We consider an application of the methodology developed within this paper which allows the simulation of upper and lower bounding processes which almost surely constrain (jump) diffusion sample paths to any specified tolerance. We demonstrate the efficacy of our approach by showing that with finite computation it is possible to determine whether or not sample paths cross various irregular barriers, simulate to any specified tolerance the first hitting time of the irregular barrier and simulate killed diffusion sample paths.