A new multifractional process with random exponent

Julien Hamonier
Université Lille
Mardi, 23 Avril, 2019 - 11:00 - 12:00
A first type of Multifractional Process with Random Exponent (MPRE) was contructed
by replacing in a wavelet series representation of Fractional Brownian Motion (FBM) the
Hurst parameter by a random variable depending on the time variable. Here, we
propose another approach for constructing another type of MPRE. In some way, this approach
is inspired by the one previously used by Surgailis. It consists in substituting
to the Hurst parameter, in a stochastic integral representation of the high-frequency part of
FBM, a random variable depending on the integration variable. The MPRE obtained in this
way offers the advantages to have a representation through classical Itô integral and to be
less difficult to simulate than the first type of MPRE. Yet, the
study of Hölder regularity of this new MPRE is a significantly more challenging problem
than in the case of the previous one. Actually, it requires to develop a new methodology
relying on an extensive use of the Haar basis.
A joint work with Antoine Ayache and Céline Esser