Stochastic Calculus with respect to Gaussian Processes: Part I

TL;DR

This paper develops a comprehensive anticipative stochastic calculus for all Gaussian processes with integral representations, extending existing methods and enabling applications in diverse fields like finance and biomedicine.

Contribution

It introduces a White Noise Theory-based stochastic calculus for a broad class of Gaussian processes, generalizing previous approaches and including processes with varying regularity.

Findings

Provides a systematic comparison with Malliavin calculus and Itô calculus.

Extends stochastic calculus to processes like multifractional Brownian motion.

Includes an Itô formula for the class of Gaussian processes.

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).