Title: 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{ô} formula) we provide here, to the ones
given by Malliavin calculus in
\cite{nualart,MV05,NuTa06,KRT07,KrRu10,LN12,SoVi14,LN12}, and by It{ô}
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).