Abstract: A first type of Multifractional Process with Random Exponent (MPRE) was constructed several years ago in [Publ. Mat. 49 (2005), 459–486] by replacing in a wavelet series representation of Fractional Brownian Motion (FBM) the Hurst parameter by a random variable depending on the time variable. In the present article, we propose another approach for constructing another type of MPRE. 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, among other things, the advantages to have a representation through classical Itô integral and to be less difficult to simulate than the first type of MPRE, previously introduced in [Publ. Mat. 49 (2005), 459–486]. 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.
Keywords: Fractional Brownian Motion, varying Hurst parameter, Haar basis, Hölder regularity, Itô integral