A new Multifractional Process with Random Exponent

Abstract: A first type of Multifractional Process with Random Exponent (MPRE) was constructed several years ago in (Ayache, Taqqu, 2005) 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 It^o integral and to be less difficult to simulate than the first type of MPRE, previously introduced in (Ayache, Taqqu, 2005). Yet, the study of H\"older 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.