Stochastic Calculus with respect to Gaussian Processes

TL;DR

This paper develops an anticipative stochastic calculus for a broad class of Gaussian processes using White Noise Theory, including definitions of stochastic integrals, Itô formulas, Tanaka formulas, and local times, generalizing existing frameworks.

Contribution

It introduces a new White Noise Theory-based stochastic calculus for Gaussian processes, encompassing fractional and multifractional Brownian motions, with generalized integrals and formulas.

Findings

Defines a stochastic integral for Gaussian processes using White Noise Theory.

Provides Itô and Tanaka formulas for Gaussian processes.

Compares the new calculus with Malliavin and Itô calculus, showing generalizations.

Abstract

Stochastic integration with respect to Gaussian processes, such as fractional Brownian motion (fBm) or multifractional Brownian motion (mBm), has raised strong interest in recent years, motivated in particular by applications in finance, Internet traffic modeling and biomedicine. The aim of this work to define and develop, using White Noise Theory, an anticipative stochastic calculus with respect to a large class of Gaussian processes, denoted G, that contains, among many other processes, Volterra processes (and thus fBm) and also mBm. This stochastic calculus includes a definition of a stochastic integral, It\^o formulas (both for tempered distributions and for functions with sub-exponential growth), a Tanaka Formula as well as a definition, and a short study, of (both weighted and non weighted) local times of elements of G . In that view, a white noise derivative of any Gaussian process G of G is defined and used to integrate, with respect to G, a large class of stochastic processes, using Wick products. A comparison of our integral wrt elements of G to the ones provided by Malliavin calculus in [AMN01] and by It\^o stochastic calculus is also made. Moreover, one shows that the stochastic calculus with respect to Gaussian processes provided in this work generalizes the stochastic calculus originally proposed for fBm in [EVdH03, BS{\O}W04, Ben03a] and for mBm in [LLV14, Leb13, LLVH14]. Likewise, it generalizes results given in [NT06] and some results given in [AMN01]. In addition, it offers alternative conditions to the ones required in [AMN01] when one deals with stochastic calculus with respect to Gaussian processes.