The XIII International Conference “Applied Stochastic Models and Data Analysis” (ASMDA-2009) June 30-July 3, 2009, Vilnius, LITHUANIA ISBN 978-9955-28-463-5 L. Sakalauskas, C. Skiadas and E. K. Zavadskas (Eds.): ASMDA-2009 Selected papers. Vilnius, 2009, pp. 400–406 © Institute of Mathematics and Informatics, 2009 © Vilnius Gediminas Technical University, 2009 400 ESTIMATION OF THE HURST INDEX OF MULTISCALE FRACTIONAL BROWNIAN MOTION WITH RANDOM OBSERVATION TIMES Jean-Marc Bardet 1 , Pierre Bertrand 2 and Medhi Fhima 3 1 SAMOS-MATISSE - UMR CNRS 8595, Université Panthéon Sorbonne, Paris I 90 rue de Tolbiac, 75013 Paris Cedex, France 2 INRIA Saclay, Parc Orsay Universit´e, 91893 Orsay Cedex, France, 2, 3 Laboratoire de Mathématiques, UMR CNRS 6620 & University Clermont-Ferrand, Campus des C´ezeaux, 63177 Aubi`ere cedex, France E-mail: 1 bardet@univ-paris1.fr; 2 Pierre.Bertrand@inria.fr; 3 Mehdi.Fhima@math.univ-bpclermont.fr Abstract: Let be a fractional Brownian motion (fBm) or a multiscale fBm. From the observation of a path of this process at random irregularly spaced times with and , its spectral density or Hurst index can be estimated. When the observation times are random but independent of the process , a standard method is the Alias Free Sampling (ASF) scheme, see Lii and Marsy, 1996. This method allows to estimate the spectral density at frequencies close to the sampling frequency when Based on the study of two real examples (Heartbeat time series and Finance), one proposes a different modeling: the spectral density is estimated on finite band of frequencies which are smaller enough than the sampling frequency, the observation times are randomly and irregularly spaced but may depend on the process itself. In this framework, an appropriated wavelet based estimator provides a con- sistent estimation of the spectral density on finite bands of frequencies. The key result is to obtain a lemma bounding the discretization error. One supports this analysis by numerical simulations. Keywords: Aliasing, Spectral density, Hurst index, Wavelet Analysis, Fractional Brownian motion, Mul- tiscale fractional Brownian motion, Finite band of frequencies, irregular time of observation. This paper is organized as follows. In the first section, one introduces modeling supported by empirical remarks. Section 2 specifies the mathematical model while Section 3 recalls some statistical results. Eventually, in Section 4, one presents numerical experiments that support our approach. 1. Introduction and Some empirical remarks To begin with, let us give an example of spectral density of heartbeat time series during working hours and sleeping hours. This data have been kindly furnished by Alain Chamoux and Gil Boudet (Clermont-Ferrand Hospital). They have recorded heartbeat series of healthy subjects. The following figures show the log-log plot of the spectral density during the 8 working hours and the 8 sleeping hours.