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Stochastic Calculus with respect to Gaussian Processes

Abstract : Stochastic integration with respect to 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 interval (that maybe infinite). This very rich, this class of Gaussian processes, which will be denoted G , contains, among many others, Volterra processes (and thus fractional Brownian motion) and multifractional Brownian motions. It also contains Gaussian processes, the regularity of which varies along the time. The stochastic calculus provided in this work allows us to derive Itô formulas, a Tanaka Formula as well as to perform a first study of (both weighted and non weighted) local times of elements of G , that includes providing occupation time formulas. A systematic comparison of the integral wrt elements of G we provide here, to the ones given by Malliavin calculus, in [AMN01], [MV05], [NT06], and by Itô stochastic calculus is also made. A complete comparison of the Itô and Tanaka formulas (resp. the results on Gaussian local times) we present here, is made with the ones proposed in [AMN01, MV05, NT06, KRT07, KR10, LN12, SV14] (resp. in [LN12]). Moreover, one shows that the stochastic calculus wrt Gaussian processes provided in this work fully generalizes and extends the ones originally proposed in [MV05] and in [NT06], as well as it generalizes some results given in [AMN01]. In particular it generalizes results given by [MV05] by allowing one to consider highly irregular processes (even more irregular that those considered in [MV05] since their regularity may vary along the time in this work). It also extends the stochastic calculus for fractional Brownian motion provided in [EVdH03, BSØW04, Ben03a] and for multifractional Brownian motion in [LLV14, Leb13, LLVH14]. In addition, the stochastic calculus presented here offers alternative conditions to the ones required in [AMN01, MV05, NT06, KRT07, KR10, LN12] and [SV14], when one deals with stochastic calculus with respect to Gaussian processes in general.
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https://hal.inria.fr/hal-01052481
Contributor : Joachim Lebovits <>
Submitted on : Thursday, February 8, 2018 - 1:33:00 PM
Last modification on : Wednesday, April 28, 2021 - 6:45:39 PM

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Joachim Lebovits. Stochastic Calculus with respect to Gaussian Processes. Potential Analysis, Springer Verlag, In press, ⟨10.1007/s11118-017-9671-5⟩. ⟨hal-01052481v3⟩

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