Generalized Multifractional Brownian Motion: Definition and Preliminary Results

Abstract

The Multifractional Brownian Motion (MBM) is a generalization of the well known Fractional Brownian Motion. One of the main reasons that makes the MBM interesting for modelization, is that one can prescribe its regularity: given any Hölder function H(t), with values in ]0, 1[, one can construct an MBM admitting at any t 0, a Hölder exponent equal to H(t 0). However, the continuity of the function H(t) is sometimes undesirable, since it restricts the field of application. In this work we define a gaussian process, called the Generalized Multifractional Brownian Motion (GMBM) that extends the MBM. This process will also depend on a functional parameter H(t) that belongs to a set 풜, but 풜 will be much more larger than the space of Hölder functions.