Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-08T17:39:11.505Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected]
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Abry, P. and Sellan, F. (1996). The wavelet-based synthesis for fractional Brownian motion proposed by F. Sellan and Y. Meyer: remarks and fast implementation. Appl. Comput. Harmon. Anal. 3, 377–383.Google Scholar
[2] Asmussen, S. and Rosiński, J. (2001). Approximations of small jumps of Lévy processes with a view towards simulation. J. Appl. Prob. 38, 482–493.Google Scholar
[3] Benassi, A., Cohen, S. and Istas, J. (2002). Identification and properties of real harmonizable fractional Lévy motions. Bernoulli 8, 97–115.Google Scholar
[4] Benassi, A., Jaffard, S. and Roux, D. (1997). Gaussian processes and pseudodifferential elliptic operators. Revista Math. Iberoamer. 13, 19–89.Google Scholar
[5] Billingsley, P. (1968). Convergence of Probability Measures. John Wiley, New York.Google Scholar
[6] Bondesson, L. (1982). On simulation from infinitely divisible distributions. Adv. Appl. Prob. 14, 855–869.Google Scholar
[7] Chan, G. and Wood, A. (1998). Simulation of multifractional Brownian motion. Proc. Comput. Statist. 233–238.Google Scholar
[8] Coeurjolly, J.-F. (2000). Simulation and identification of the fractional Brownian motion: a bibliographical and comparative study. J. Statist. Software 5, No. 7. Available at http://www.jstatsoft.org/.Google Scholar
[9] Dury, M. E. (2001). Identification et simulation d'une classe de processus stables autosimilaires à accroissements stationnaires. Doctoral Thesis, Université Blaise Pascal, Clermont-Ferrand.Google Scholar
[10] Haagerup, U. (1981). The best constants in the Khintchine inequality. Studia Math. 70, 231–283.CrossRefGoogle Scholar
[11] Lacaux, C. (2004). Real harmonizable multifractional Lévy motions. To appear in Ann. Inst. H. Poincaré Prob. Statist.Google Scholar
[12] Ledoux, M. and Talagrand, M. (1991). Probability in Banach Spaces (Results Math. Relat. Areas (3) 23). Springer, Berlin.CrossRefGoogle Scholar
[13] Mallat, S. (1989). A theory for multiresolution signal decomposition: the wavelet representation. IEEE Trans. Pattern Anal. Mach. Intellig. 11, 674–693.Google Scholar
[14] Mandelbrot, B. and Ness, J. V. (1968). Fractional Brownian motion, fractional noises and applications. SIAM Rev. 10, 422–437.CrossRefGoogle Scholar
[15] Peltier, R. and Lévy Véhel, J. (1996). Multifractional Brownian motion: definition and preliminary results. Tech. Rep. RR-2645, INRIA. Available at http://www-syntim.inria.fr/fractales/.Google Scholar
[16] Petrov, V. V. (1995). Limit Theorems of Probability Theory (Oxford Studies Prob. 4). Oxford University Press.Google Scholar
[17] Rosiński, J., (1990). On series representations of infinitely divisible random vectors. Ann. Prob. 18, 405–430.Google Scholar
[18] Rosiński, J., (2001). Series representations of Lévy processes from the perspective of point processes. In Lévy Processes, Birkhäuser, Boston, MA, pp. 401–415.Google Scholar
[19] Rubenthaler, S. (2003). Numerical simulation of the solution of a stochastic differential equation driven by a Lévy process. Stoch. Process. Appl. 103, 311–349.CrossRefGoogle Scholar
[20] Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes. Chapman and Hall, New York.Google Scholar
[21] Vidakovic, B. (1999). Statistical Modeling by Wavelets. John Wiley, New York.Google Scholar
[22] Wiktorsson, M. (2002). Simulation of stochastic integrals with respect to Lévy processes of type G. Stoch. Process. Appl. 101, 113–125.CrossRefGoogle Scholar
[23] Wood, A. T. A. and Chan, G. (1994). Simulation of stationary Gaussian processes in [0,1]d . J. Comput. Graph. Statist. 3, 409–432.Google Scholar