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Series representation and simulation of multifractional Lévy motions

  • Céline Lacaux (a1)

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 X h is split into two independent RHMLMs, X ε,1 and X ε,2. More precisely, X ε,2 is an RHMLM whose control measure is finite. It can then be rewritten as a generalized shot-noise series. The asymptotic behaviour of X ε,1 as ε → 0+ is further elaborated. Sufficient conditions to approximate X ε,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.

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Corresponding author

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

References

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[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, 377383.
[2] Asmussen, S. and Rosiński, J. (2001). Approximations of small jumps of Lévy processes with a view towards simulation. J. Appl. Prob. 38, 482493.
[3] Benassi, A., Cohen, S. and Istas, J. (2002). Identification and properties of real harmonizable fractional Lévy motions. Bernoulli 8, 97115.
[4] Benassi, A., Jaffard, S. and Roux, D. (1997). Gaussian processes and pseudodifferential elliptic operators. Revista Math. Iberoamer. 13, 1989.
[5] Billingsley, P. (1968). Convergence of Probability Measures. John Wiley, New York.
[6] Bondesson, L. (1982). On simulation from infinitely divisible distributions. Adv. Appl. Prob. 14, 855869.
[7] Chan, G. and Wood, A. (1998). Simulation of multifractional Brownian motion. Proc. Comput. Statist. 233238.
[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/.
[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.
[10] Haagerup, U. (1981). The best constants in the Khintchine inequality. Studia Math. 70, 231283.
[11] Lacaux, C. (2004). Real harmonizable multifractional Lévy motions. To appear in Ann. Inst. H. Poincaré Prob. Statist.
[12] Ledoux, M. and Talagrand, M. (1991). Probability in Banach Spaces (Results Math. Relat. Areas (3) 23). Springer, Berlin.
[13] Mallat, S. (1989). A theory for multiresolution signal decomposition: the wavelet representation. IEEE Trans. Pattern Anal. Mach. Intellig. 11, 674693.
[14] Mandelbrot, B. and Ness, J. V. (1968). Fractional Brownian motion, fractional noises and applications. SIAM Rev. 10, 422437.
[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/.
[16] Petrov, V. V. (1995). Limit Theorems of Probability Theory (Oxford Studies Prob. 4). Oxford University Press.
[17] Rosiński, J., (1990). On series representations of infinitely divisible random vectors. Ann. Prob. 18, 405430.
[18] Rosiński, J., (2001). Series representations of Lévy processes from the perspective of point processes. In Lévy Processes, Birkhäuser, Boston, MA, pp. 401415.
[19] Rubenthaler, S. (2003). Numerical simulation of the solution of a stochastic differential equation driven by a Lévy process. Stoch. Process. Appl. 103, 311349.
[20] Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes. Chapman and Hall, New York.
[21] Vidakovic, B. (1999). Statistical Modeling by Wavelets. John Wiley, New York.
[22] Wiktorsson, M. (2002). Simulation of stochastic integrals with respect to Lévy processes of type G. Stoch. Process. Appl. 101, 113125.
[23] Wood, A. T. A. and Chan, G. (1994). Simulation of stationary Gaussian processes in [0,1]d . J. Comput. Graph. Statist. 3, 409432.

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Series representation and simulation of multifractional Lévy motions

  • Céline Lacaux (a1)

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