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

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.