From Stochastic Integration wrt Fractional Brownian Motion to Stochastic Integration wrt Multifractional Brownian Motion

TL;DR

This paper explores how to define stochastic integrals with respect to multifractional Brownian motion by approximating it with fractional Brownian motions and extending existing integration methods.

Contribution

It introduces a novel approach to define stochastic integration with respect to mBm using approximation by fBm and White Noise Theory.

Findings

mBm can be approximated by a sequence of fBms in law

Stochastic integrals w.r.t. mBm can be constructed from fBm integrals

The method extends stochastic calculus to more general Gaussian processes

Abstract

Stochastic integration w.r.t. fractional Brownian motion (fBm) has raised strong interest in recent years, motivated in particular by applications in finance and Internet traffic modelling. Since fBm is not a semi-martingale, stochastic integration requires specific developments. Multifractional Brownian motion (mBm) is a Gaussian process that generalizes fBm by letting the local H\"older exponent vary in time. This is useful in various areas, including financial modelling and biomedicine. In this work we start from the fact, established in \cite[Thm 2.1.(i)]{fBm_to_mBm_HerbinLebovitsVehel}, that an mBm may be approximated, in law, by a sequence of "tangent" fBms. We used this result to show how one can define a stochastic integral w.r.t. mBm from the stochastic integral w.r.t. fBm, defined in \cite{Ben1}, in the White Noise Theory sense.