We construct Generalized Multifractional Processes with Random Exponent (GMPREs). These processes, defined through a wavelet representation, are obtained by replacing the Hurst parameter of Fractional Brownian Motion by a sequence of continuous random processes. We show that these GMPREs can have the most general pointwise H#x00F6;lder exponent function possible, namely, a random H#x00F6;lder exponent which is a function of time and which can be expressed in the strong sense (almost surely for all ), as a of an arbitrary sequence of continuous processes with values in .
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Antoine Ayache, Stéphane Jaffard, Murad S. Taqqu, Wavelet construction of Generalized Multifractional processes. Rev. Mat. Iberoam. 23 (2007), no. 1, pp. 327–370