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 $t$), as a $\liminf$ of an arbitrary sequence of continuous processes with values in $[0,1]$.
"Wavelet construction of Generalized Multifractional processes." Rev. Mat. Iberoamericana 23 (1) 327 - 370, April, 2007.