Stochastic Calculus with respect to multifractional Brownian motion

Abstract

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. Itô 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 Itô 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-Itô integral, Itô formula, Tanaka formula, Stochastic differential equations.