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Stochastic Calculus with Respect to Gaussian Processes

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Abstract

Stochastic integration 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 Itô formula) we provide here, to the ones given by Malliavin calculus in Aloś (Ann. Probab. 29(2), 766–801 2001), Mocioalca and Viens (J. Funct. Anal. 222(2), 385–434 2005), Nualart and Taqqu (Stoch. Anal Appl. 24(3), 599–614 2006), Kruk et al. (J. Funct. Anal. 249(1), 92–142 2007), Kruk and Russo (2010), Lei and Nualart (Commun. Stoch. Anal. 6(3), 379–402 2012) and Sottinen and Viitasaari (2014), and by Itô stochastic calculus is also made. Not only our stochastic calculus fully generalizes and extends the ones originally proposed in Mocioalca and Viens (J. Funct. Anal. 222(2), 385–434 2005) and in Nualart and Taqqu (Stoch. Anal Appl. 24(3), 599–614 2006) for Gaussian processes, but also the ones proposed in Bender (Stoch. Process. Appl. 104, 81–106 2003), Biagini et al. (2004) and Elliott and Van der Hoek (Math. Financ. 13(2), 301–330 2003) for fractional Brownian motion (resp. in Lebovits, Ann. Univ. Buchar. Math. Ser. 4(LXII)(1), 397–413 2013; Lebovits and Lévy Véhel Stoch. Int. J. Probab. Stoch. Processes 86(1), 87–124 2014; Lebovits et al. Stoch. Process. Appl. 124(1), 678–708 2014 for multifractional Brownian motion).

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Acknowledgments

I want to express my deep gratitude to Jacques Lévy Véhel for his advices and for the very stimulating discussions we had about this work. I also want to thank Professor T. Hida for his warm welcome at the University of Nagoya, where a part of this paper was written, as well as Professor L. Chen and the Institute for Mathematical Sciences of Singapore (NUS), where another part of this paper was written. I also thanks the Associate Editor as well as the anonymous referee for his remarks that greatly improve the quality of this paper and especially Section 4.2.

This work is dedicated to the memory of Professor Marc Yor.

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  1. Laboratoire Analyse, Géométrie et Applications, C.N.R.S. (UMR 7539), Université Paris 13, Sorbonne Paris Cité, 99 avenue Jean-Baptiste Clément, 93430, Villetaneuse, France

    Joachim Lebovits

  2. MAThematic Center of Heidelberg, University of Heidelberg, INF 294, 69120, Heidelberg, Germany

    Joachim Lebovits

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Lebovits, J. Stochastic Calculus with Respect to Gaussian Processes. Potential Anal 50, 1–42 (2019). https://doi.org/10.1007/s11118-017-9671-5

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