  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.
