Implementable coupling of Lévy process and Brownian motion

Vladimir Fomichov; Jorge González Cázares; Jevgenijs Ivanovs

doi:10.1016/j.spa.2021.09.008

Implementable coupling of Lévy process and Brownian motion

Vladimir Fomichov, Jorge González Cázares^*, Jevgenijs Ivanovs

^*Corresponding author for this work

Department of Mathematics

Research output: Contribution to journal/Conference contribution in journal/Contribution to newspaper › Journal article › Research › peer-review

Abstract

We provide a simple algorithm for construction of Brownian paths approximating those of a Lévy process on a finite time interval. It requires knowledge of the Lévy 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.

Original language	English
Journal	Stochastic Processes and Their Applications
Volume	142
Pages (from-to)	407-431
ISSN	0304-4149
DOIs	https://doi.org/10.1016/j.spa.2021.09.008
Publication status	Published - Dec 2021

Keywords

Approximation
Distributional model risk
Lévy processes
Multilevel Monte Carlo
Wasserstein distance

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10.1016/j.spa.2021.09.008

https://arxiv.org/pdf/2103.11475.pdf

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title = "Implementable coupling of L{\'e}vy process and Brownian motion",

abstract = "We provide a simple algorithm for construction of Brownian paths approximating those of a L{\'e}vy process on a finite time interval. It requires knowledge of the L{\'e}vy 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.",

keywords = "Approximation, Distributional model risk, L{\'e}vy processes, Multilevel Monte Carlo, Wasserstein distance",

author = "Vladimir Fomichov and {Gonz{\'a}lez C{\'a}zares}, Jorge and Jevgenijs Ivanovs",

note = "Publisher Copyright: {\textcopyright} 2021 Elsevier B.V.",

year = "2021",

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doi = "10.1016/j.spa.2021.09.008",

language = "English",

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TY - JOUR

T1 - Implementable coupling of Lévy process and Brownian motion

AU - Fomichov, Vladimir

AU - González Cázares, Jorge

AU - Ivanovs, Jevgenijs

PY - 2021/12

Y1 - 2021/12

N2 - We provide a simple algorithm for construction of Brownian paths approximating those of a Lévy process on a finite time interval. It requires knowledge of the Lévy 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.

AB - We provide a simple algorithm for construction of Brownian paths approximating those of a Lévy process on a finite time interval. It requires knowledge of the Lévy 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.

KW - Approximation

KW - Distributional model risk

KW - Lévy processes

KW - Multilevel Monte Carlo

KW - Wasserstein distance

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U2 - 10.1016/j.spa.2021.09.008

DO - 10.1016/j.spa.2021.09.008

M3 - Journal article

AN - SCOPUS:85115889505

SN - 0304-4149

VL - 142

SP - 407

EP - 431

JO - Stochastic Processes and Their Applications

JF - Stochastic Processes and Their Applications

ER -

Implementable coupling of Lévy process and Brownian motion

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