A new multifractional process with random exponent

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