Jacques Lévy Véhel

White noise-based stochastic calculus with respect to multifractional Brownian motion

Stochastic calculus with respect to fractional Brownian motion (fBm) has attracted a lot of interest in recent years, motivated in particular by applications in finance and Internet traffic modeling. Multifractional Brownian motion (mBm) is an extension of fBm enabling to control the local regularity of the process. It is obtained by replacing the constant Hurst parameter H of fBm by a function h(t), thus allowing for a finer modelling of various phenomena. In this work we extend to mBm the construction of the Wick–Itô stochastic integral with respect to fBm, as originally proposed in Bender (Stoch. Process. Appl. 104 (2003), pp. 81–106), Bender (Bernouilli 9(6) (2003), pp. 955–983), Biagini et al. (Proceedings of Royal Society, special issue on stochastic analysis and applications, 2004, pp. 347–372) and Elliott and Van der Hoek (Math. Finance 13(2) (2003), pp. 301–330). In that view, a multifractional white noise is defined and used to integrate with respect to mBm a large class of stochastic processes using Wick products. Itô formulas (both for tempered distributions and for functions with sub-exponential growth) are obtained, as well as a Tanaka Formula.

Self-regulating processes-based modeling for arrhythmia characterization

We propose a new model for RR interval records based on the empirical finding that the amplitude of RR records is negatively correlated with its regularity. We develop the mathematics needed to define rigorously this model, as well as the statistical tools necessary for estimating the parameters of the model on sampled data. We use this to analyze finely the impact of ventricular extrasystole of RR records, and to assess the efficiency of certain treatments on this pathology.