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Series representation and simulation of multifractional Lévy motions

Part of: Probabilistic methods, simulation and stochastic differential equations Distribution theory - Probability Stochastic processes Limit theorems

Published online by Cambridge University Press:  01 July 2016

Céline Lacaux
Céline Lacaux*
Affiliation:
Université Paul Sabatier, Toulouse
*
∗ Postal address: Université Paul Sabatier, UFR MIG, Laboratoire de Statistique et Probabilités, 118, Route de Narbonne, 31062 Toulouse, France. Email address: celine.lacaux@math.ups-tlse.fr
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Abstract

This paper introduces a method of generating real harmonizable multifractional Lévy motions (RHMLMs). The simulation of these fields is closely related to that of infinitely divisible laws or Lévy processes. In the case where the control measure of the RHMLM is finite, generalized shot-noise series are used. An estimation of the error is also given. Otherwise, the RHMLM Xh is split into two independent RHMLMs, Xε,1 and Xε,2. More precisely, Xε,2 is an RHMLM whose control measure is finite. It can then be rewritten as a generalized shot-noise series. The asymptotic behaviour of Xε,1 as ε → 0+ is further elaborated. Sufficient conditions to approximate Xε,1 by a multifractional Brownian motion are given. The error rate in terms of Berry-Esseen bounds is then discussed. Finally, some examples of simulation are given.

Keywords

Functional central limit theoremgeneralized shot-noise seriesinfinitely divisible distributionmultifractional Brownian motionsimulation

MSC classification

Primary: 60E07: Infinitely divisible distributions; stable distributions
Secondary: 65C99: None of the above, but in this section 60F05: Central limit and other weak theorems 60F17: Functional limit theorems; invariance principles 60G12: General second-order processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 36 , Issue 1 , March 2004 , pp. 171 - 197
Copyright
Copyright © Applied Probability Trust 2004 

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