Implementable coupling of L\'evy process and Brownian motion

TL;DR

This paper introduces a simple, recursive algorithm for approximating Lévy process paths with Brownian motion, based on reordering increments, with proven error bounds and applications in risk modeling and Monte Carlo simulations.

Contribution

It presents a novel, implementable coupling method that approximates Lévy paths using Brownian motion through reordering of increments, with theoretical error analysis.

Findings

Provides an upper bound on mean squared maximal distance between paths

Determines suitable mesh size in various regimes

Demonstrates applications in risk modeling and Monte Carlo methods

Abstract

We provide a simple algorithm for construction of Brownian paths approximating those of a L\'evy process on a finite time interval. It requires knowledge of the L\'evy process trajectory on a chosen regular grid and the law of its endpoint, or the ability to simulate from that. This algorithm is based on reordering of Brownian increments, and it can be applied in a recursive manner. We establish an upper bound on the mean squared maximal distance between the paths and determine a suitable mesh size in various asymptotic regimes. The analysis proceeds by reduction to the comonotonic coupling of increments. Applications to model risk and multilevel Monte Carlo are discussed in detail, and numerical examples are provided.