Simulation of stochastic volterra equations driven by space-time Lévy noise

Bohan Chen; Carsten Chong; Claudia Klüppelberg

doi:10.1007/978-3-319-25826-3_10

Simulation of stochastic volterra equations driven by space-time Lévy noise

Bohan Chen, Carsten Chong, Claudia Klüppelberg^*

^*Corresponding author for this work

Research output: Chapter in Book/Conference Proceeding/Report › Book Chapter › peer-review

6 Citations (Scopus)

Abstract

In this paper we investigate two numerical schemes for the simulation of stochastic Volterra equations driven by space-time Lévy noise of pure-jump type. The first one is based on truncating the small jumps of the noise, while the second one relies on series representation techniques for infinitely divisible random variables. Under reasonable assumptions, we prove for both methods L^p- and almost sure convergence of the approximations to the true solution of the Volterra equation. We give explicit convergence rates in terms of the Volterra kernel and the characteristics of the noise. A simulation study visualizes the most important path properties of the investigated processes.

Original language	English
Title of host publication	The Fascination of Probability, Statistics and their Applications
Subtitle of host publication	In Honour of Ole E. Barndorff-Nielsen
Publisher	Springer International Publishing
Pages	209-229
Number of pages	21
ISBN (Electronic)	9783319258263
ISBN (Print)	9783319258249
DOIs	https://doi.org/10.1007/978-3-319-25826-3_10
Publication status	Published - 1 Jan 2015
Externally published	Yes

Bibliographical note

Publisher Copyright:
© Springer International Publishing Switzerland 2016.

Keywords

Simulation of SPDEs
Simulation of stochastic Volterra equations
Space-time lévy noise
Stochastic heat equation
Stochastic partial differential equation

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10.1007/978-3-319-25826-3_10

Cite this

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title = "Simulation of stochastic volterra equations driven by space-time L{\'e}vy noise",

abstract = "In this paper we investigate two numerical schemes for the simulation of stochastic Volterra equations driven by space-time L{\'e}vy noise of pure-jump type. The first one is based on truncating the small jumps of the noise, while the second one relies on series representation techniques for infinitely divisible random variables. Under reasonable assumptions, we prove for both methods Lp- and almost sure convergence of the approximations to the true solution of the Volterra equation. We give explicit convergence rates in terms of the Volterra kernel and the characteristics of the noise. A simulation study visualizes the most important path properties of the investigated processes.",

keywords = "Simulation of SPDEs, Simulation of stochastic Volterra equations, Space-time l{\'e}vy noise, Stochastic heat equation, Stochastic partial differential equation",

author = "Bohan Chen and Carsten Chong and Claudia Kl{\"u}ppelberg",

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doi = "10.1007/978-3-319-25826-3\_10",

language = "English",

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Simulation of stochastic volterra equations driven by space-time Lévy noise. / Chen, Bohan; Chong, Carsten; Klüppelberg, Claudia.
The Fascination of Probability, Statistics and their Applications: In Honour of Ole E. Barndorff-Nielsen. Springer International Publishing, 2015. p. 209-229.

Research output: Chapter in Book/Conference Proceeding/Report › Book Chapter › peer-review

TY - CHAP

T1 - Simulation of stochastic volterra equations driven by space-time Lévy noise

AU - Chen, Bohan

AU - Chong, Carsten

AU - Klüppelberg, Claudia

N1 - Publisher Copyright: © Springer International Publishing Switzerland 2016.

PY - 2015/1/1

Y1 - 2015/1/1

N2 - In this paper we investigate two numerical schemes for the simulation of stochastic Volterra equations driven by space-time Lévy noise of pure-jump type. The first one is based on truncating the small jumps of the noise, while the second one relies on series representation techniques for infinitely divisible random variables. Under reasonable assumptions, we prove for both methods Lp- and almost sure convergence of the approximations to the true solution of the Volterra equation. We give explicit convergence rates in terms of the Volterra kernel and the characteristics of the noise. A simulation study visualizes the most important path properties of the investigated processes.

AB - In this paper we investigate two numerical schemes for the simulation of stochastic Volterra equations driven by space-time Lévy noise of pure-jump type. The first one is based on truncating the small jumps of the noise, while the second one relies on series representation techniques for infinitely divisible random variables. Under reasonable assumptions, we prove for both methods Lp- and almost sure convergence of the approximations to the true solution of the Volterra equation. We give explicit convergence rates in terms of the Volterra kernel and the characteristics of the noise. A simulation study visualizes the most important path properties of the investigated processes.

KW - Simulation of SPDEs

KW - Simulation of stochastic Volterra equations

KW - Space-time lévy noise

KW - Stochastic heat equation

KW - Stochastic partial differential equation

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UR - https://www.scopus.com/pages/publications/85012119065

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DO - 10.1007/978-3-319-25826-3_10

M3 - Book Chapter

SN - 9783319258249

SP - 209

EP - 229

BT - The Fascination of Probability, Statistics and their Applications

PB - Springer International Publishing

ER -

Simulation of stochastic volterra equations driven by space-time Lévy noise

Abstract

Bibliographical note

Keywords

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