Stochastic Calculus with respect to Gaussian Processes: Part I

Abstract

Stochastic integration \textit{wrt} Gaussian processes has raised strong interest in recent years, motivated in particular by its applications in Internet traffic modeling, biomedicine and finance. The aim of this work is to define and develop a White Noise Theory-based anticipative stochastic calculus with respect to all Gaussian processes that have an integral representation over a real (maybe infinite) interval. Very rich, this class of Gaussian processes contains, among many others, Volterra processes (and thus fractional Brownian motion) as well as processes the regularity of which varies along the time (such as multifractional Brownian motion).A systematic comparison of the stochastic calculus (including It{\^o} formula) we provide here, to the ones given by Malliavin calculus in \cite{nualart,MV05,NuTa06,KRT07,KrRu10,LN12,SoVi14,LN12}, and by It{\^o} stochastic calculus is also made. Not only our stochastic calculus fully generalizes and extends the ones originally proposed in \cite{MV05} and in \cite{NuTa06} for Gaussian processes, but also the ones proposed in \cite{ell,bosw,ben1} for fractional Brownian motion (\textit{resp.} in \cite{JLJLV1,JL13,LLVH} for multifractional Brownian motion).