  Stochastic calculus with respect to fractional Brownian motion (fBm) has
attracted a lot of interest in recent years, motivated in particular by
applications in finance and Internet traffic modeling. Multifractional Brownian
motion (mBm) is a Gaussian extension of fBm that allows to control the
pointwise regularity of the paths of the process and to decouple it from its
long range dependence properties. This generalization is obtained by replacing
the constant Hurst parameter H of fBm by a function h(t). Multifractional
Brownian motion has proved useful in many applications, including the ones just
mentioned. In this work we extend to mBm the construction of a stochastic
integral with respect to fBm. This stochastic integral is based on white noise
theory, as originally proposed in [15], [6], [4] and in [5]. In that view, a
multifractional white noise is defined, which allows to integrate with respect
to mBm a large class of stochastic processes using Wick products. It\^o
formulas (both for tempered distributions and for functions with
sub-exponential growth) are given, along with a Tanaka Formula. The cases of
two specific functions h which give notable It\^o formulas are presented. We
also study a stochastic differential equation driven by a mBm. Keywords:
multifractional Brownian motion, Gaussian processes, White Noise Theory,
S-Transform, Wick-It\^o integral, It\^o formula, Tanaka formula, Stochastic
differential equations.
