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).
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Alòs, E., Mazet, O., Nualart, D.: Stochastic calculus with respect to Gaussian processes. Ann. Probab. 29(2), 766–801 (2001)
Bally, V.: An elementary introduction to Malliavin calculus. Rapport de recherche no 4718 INRIA (2003)
Boufoussi, B., Dozzi, M., Marty, R.: Local time and Tanaka formula for a Volterra-type multifractional Gaussian process. Bernoulli 16(4), 1294–1311 (2010)
Bender, C.: An Itô formula for generalized functionals of a fractional Brownian motion with arbitrary Hurst parameter. Stoch. Process. Appl. 104, 81–106 (2003)
Bender, C.: An S-transform approach to integration with respect to a fractional Brownian motion. Bernoulli 9(6), 955–983 (2003)
Benassi, A., Jaffard, S., Roux, D.: Elliptic Gaussian random processes. Rev. Mat. Iberoamericana 13(1), 19–90 (1997)
Biagini, F., Sulem, A., Øksendal, B., Wallner, N.N.: An introduction to white-noise theory and Malliavin calculus for fractional Brownian motion. In: Proc. Royal Society, Special Issue on Stochastic Analysis and Applications, pp. 347–372 (2004)
Corlay, S., Lebovits, J., Lévy Véhel, J.: Multifractional stochastic volatility models. Math. Financ. 24(2), 364–402 (2014)
Cheridito, P., Nualart, D.: Stochastic integral of divergence type with respect to fractional Brownian motion with Hurst parameter \(H\in (0,{1\over 2})\). Ann. Inst. H Poincaré Probab. Statist. 41(6), 1049–1081 (2005)
Coutin, L.: An introduction to (stochastic) calculus with respect to fractional Brownian motion. Séminaire de Probabilités XL 1899, 3–65 (2007)
Elliott, R.J., Van der Hoek, J.: A general fractional white noise theory and applications to finance. Math. Financ. 13(2), 301–330 (2003)
Föllmer, H.: Calcul d’Itô sans probabilités. In: Seminar on Probability, XV (Univ. Strasbourg, Strasbourg, 1979/1980) (French), Volume 850 of Lecture Notes in Math., pp. 143–150. Springer, Berlin (1981)
Friz, P. K., Victoir, N.B.: Multidimensional Stochastic Processes as Rough Paths: Theory and Applications. Cambridge University Press (2010)
Gasbarra, D., Sottinen, T., Valkeila, E.: Gaussian bridges. In: Stochastic Analysis and Applications, Volume 2 of Abel Symp., pp. 361–382. Springer, Berlin (2007)
Gaveau, B., Trauber, P.: L’intégrale stochastique comme opérateur de divergence dans l’espace fonctionnel. J. Funct. Anal. 46(2), 230–238 (1982)
Hida, T., Hitsuda, M.: Gaussian Processes, Volume 120 of Translations of Mathematical Monographs. American Mathematical Society, Providence (1993). Translated from the 1976 Japanese original by the authors
Hida, T.: Analysis of Brownian Functionals. Carleton Univ., Ottawa (1975). Carleton Mathematical Lecture Notes No. 13
Hitsuda, M.: Formula for Brownian partial derivatives. Second Japan-USSR Symposium on Probability Theory, pp. 111–114. Kyoto (1972)
Hitsuda, M.: Formula for Brownian partial derivatives. In: Proceedings of Faculty of Integrated Arts and Sciences. Hiroshima University III-4, pp. 1–15 (1978)
Hida, T., Kuo, H., Potthoff, J., Streit, L.: White Noise. An Infinite Dimensional Calculus, vol. 253. Kluwer Academic Publishers (1993)
Holden, H., Oksendal, B., Ubøe, J., Zhang, T.: Stochastic Partial Differential Equations. A Modeling. White Noise Functional Approach, 2nd edn. Springer (2010)
Hille, E., Phillips, R.S.: Functional Analysis and Semi-Groups, vol. 31. American Mathematical Society (1957)
Hida, T., Si, S: An Innovation Approach to Random Fields. World Scientific Publishing Co. Inc, River Edge (2004). Application of white noise theory
Houdré, C., Villa, J.: An example of infinite dimensional quasi-helix. In: Stochastic models (Mexico City, 2002), Volume 336 of Contemp. Math., pp. 195–201. Amer. Math. Soc., Providence (2003)
Janson, S.: Gaussian Hilbert Spaces, Volume 129 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (1997)
Kolmogorov, A.: Wienersche Spiralen und einige andere interessante Kurven in Hilbertsche Raum. C. R. (Dokl.) Acad. Sci. URSS 26, 115–118 (1940)
Kruk, I., Russo, F.: Malliavin Skorohod calculus and Paley-Wiener integral for covariance singular processes. Preprint (2010)
Kruk, I., Russo, F., Tudor, C.A.: Wiener integrals, Malliavin calculus and covariance measure structure. J. Funct. Anal. 249(1), 92–142 (2007)
Kubo, I.: Itô formula for generalized Brownian functionals. In: Theory and Application of Random Fields (Bangalore, 1982), Volume 49 of Lect. Notes Control Inf. Sci., pp. 156–166. Springer, Berlin (1983)
Kuo, H.H.: White Noise Distribution Theory. CRC-Press (1996)
Lebovits, J.: From stochastic integral w.r.t. fractional Brownian motion to stochastic integral w.r.t. multifractional Brownian motion. Ann. Univ. Buchar. Math. Ser. 4(LXII)(1), 397–413 (2013)
Lebovits, J.: Local Times of Gaussian Processes: Stochastic Calculus wrt Gaussian Processes Part II. Preprint (2017)
Lebovits, J., Lévy Véhel, J.: White noise-based stochastic calculus with respect to multifractional Brownian motion. Stochastics Int. J. Probab. Stoch. Processes 86(1), 87–124 (2014)
Lebovits, J., Lévy Véhel, J., Herbin, E.: Stochastic integration with respect to Brownian motion via tangent fractional multifractional Brownian motions. Stochastic Process. Appl. 124(1), 678–708 (2014)
León, J.A., Nualart, D.: An extension of the divergence operator for Gaussian processes. Stochastic Process. Appl. 115(3), 481–492 (2005)
Lei, P., Nualart, D.: Stochastic calculus for Gaussian processes and application to hitting times. Commun. Stoch. Anal. 6(3), 379–402 (2012)
Mishura, Y.S.: Stochastic Calculus for Fractional Brownian Motion and Related Processes, Volume 1929 of Lecture Notes in Mathematics. Springer-Verlag, Berlin (2008)
Mocioalca, O., Viens, F.: Skorohod integration and stochastic calculus beyond the fractional Brownian scale. J. Funct. Anal. 222(2), 385–434 (2005)
Mandelbrot, B., Van Ness, J.W.: Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10, 422–437 (1968)
Mansuy, R., Yor, M.: Random Times and Enlargements of Filtrations in a Brownian setting. Springer-Verlag New York Inc. (2006)
Nualart, D., Taqqu, M.S.: Wick-Itô formula for Gaussian processes. Stoch. Anal Appl. 24(3), 599–614 (2006)
Nualart, D.: A white noise approach to fractional Brownian motion. In: Stochastic Analysis: Classical and Quantum, pp. 112–126. World Sci. Publ., Hackensack (2005)
Nualart, D.: The Malliavin Calculus and Related Topics. Springer (2006)
Peltier, R., Lévy Véhel, J.: Multifractional Brownian motion: definition and preliminary results, 1995. rapport de recherche de l’INRIA n 0(2645)
Russo, F., Vallois, P.: Elements of stochastic calculus via regularization. In: Séminaire de Probabilités XL, Volume 1899 of Lecture Notes in Math., pp. 147–185. Springer, Berlin (2007)
Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion, Volume 293 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 3rd edn. Springer-Verlag, Berlin (1999)
Si, S.: Introduction to Hida Distributions. World Scientific Publishing Co. Pte. Ltd., Hackensack (2012)
Skorohod, A.V.: On a generalization of the stochastic integral. Teor. Verojatnost. i Primenen. 20(2), 223–238 (1975)
Stricker, C.: Semimartingales gaussiennes—application au problème de l’innovation. Z. Wahrsch. Verw. Gebiete 64(3), 303–312 (1983)
Sottinen, T., Viitassaari, L.: Pathwise integrals and Itô-Tanaka formula for Gaussian processes. Preprint arXiv:1307.3578 (2013)
Sottinen, T., Viitasaari, L.: Stochastic Analysis of Gaussian Processes via Fredholm Representation Preprint, arXiv:1410.2230 (2014)
Thangavelu, S.: Lectures of Hermite and Laguerre Expansions. Princeton University Press (1993)
Widder, D.V.: Positive temperatures on an infinite rod. Trans. Amer. Math. Soc. 55, 85–95 (1944)
Young, L.C.: An inequality of the Hölder type, connected with Stieltjes integration. Acta Math. 67(1), 251–282 (1936)
Zähle, M.: Integration with respect to fractal functions and stochastic calculus I. Probab. Theory Related Fields 111(3), 333–374 (1998)
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Lebovits, J. Stochastic Calculus with Respect to Gaussian Processes. Potential Anal 50, 1–42 (2019). https://doi.org/10.1007/s11118-017-9671-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11118-017-9671-5
Keywords
- Stochastic analysis
- White noise theory
- Gaussian processes
- Wick-Itô integrals
- Itô formula
- Varying regularity processes