Joachim Lebovits

(Université Paris 13)

(Université Paris 13)

Thiele Seminar

Thursday, 5 March, 2015, at 13:15-14:00, in Koll. D (1531-211)

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, Itô 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 [1] and by Itô 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 [4, 3, 2] and for mBm in [6, 5, 7]. Likewise, it generalizes results given in [8] and some results given in [1]. In addition, it offers alternative conditions to the ones required in [1] when one deals with stochastic calculus with respect to Gaussian processes.

Key words: Gaussian processes, Stochastic Analysis, White noise theory, Wick-Itô and Hitsuda-Skorohod integrals, Itô and Tanaka formulas, fractional and multifractional Brownian motions.

References

[1] E. Alòs, O. Mazet, and D. Nualart. Stochastic calculus with respect to Gaussian processes. Ann. Probab., 29(2):766–801, 2001.

[2] C. Bender. An Itô formula for generalized functionals of a fractional Brownian motion with arbitrary Hurst parameter. Stochastic Processes and their Applications, 104:81–106, 2003.

[3] F. Biagini, A. Sulem, B. Øksendal, and N.N. Wallner. An introduction to white-noise theory and Malliavin calculus for fractional Brownian motion. Proc. Royal Society, special issue on stochastic analysis and applications, pages 347–372, 2004.

[4] R.J. Elliott and J. Van der Hoek. A general fractional white noise theory and applications to finance. Mathematical Finance, 13(2):301–330, 2003.

[5] J. Lebovits. 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.

[6] J. Lebovits and J. Lévy Véhel. White noise-based stochastic calculus with respect to multifractional Brownian motion. Stochastics An International Journal of Probability and Stochastic Processes, 86(1):87–124, 2014.

[7] J. Lebovits, J. Lévy Véhel, and E. Herbin. Stochastic integration with respect to multifractional Brownian motion via tangent fractional Brownian motions. Stochastic Process. Appl., 124(1):678–708, 2014.

[8] D. Nualart and M.S. Taqqu. Wick-Itô formula for Gaussian processes. Stoch. Anal. Appl., 24(3):599–614, 2006.

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 [1] and by Itô 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 [4, 3, 2] and for mBm in [6, 5, 7]. Likewise, it generalizes results given in [8] and some results given in [1]. In addition, it offers alternative conditions to the ones required in [1] when one deals with stochastic calculus with respect to Gaussian processes.

Key words: Gaussian processes, Stochastic Analysis, White noise theory, Wick-Itô and Hitsuda-Skorohod integrals, Itô and Tanaka formulas, fractional and multifractional Brownian motions.

References

[1] E. Alòs, O. Mazet, and D. Nualart. Stochastic calculus with respect to Gaussian processes. Ann. Probab., 29(2):766–801, 2001.

[2] C. Bender. An Itô formula for generalized functionals of a fractional Brownian motion with arbitrary Hurst parameter. Stochastic Processes and their Applications, 104:81–106, 2003.

[3] F. Biagini, A. Sulem, B. Øksendal, and N.N. Wallner. An introduction to white-noise theory and Malliavin calculus for fractional Brownian motion. Proc. Royal Society, special issue on stochastic analysis and applications, pages 347–372, 2004.

[4] R.J. Elliott and J. Van der Hoek. A general fractional white noise theory and applications to finance. Mathematical Finance, 13(2):301–330, 2003.

[5] J. Lebovits. 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.

[6] J. Lebovits and J. Lévy Véhel. White noise-based stochastic calculus with respect to multifractional Brownian motion. Stochastics An International Journal of Probability and Stochastic Processes, 86(1):87–124, 2014.

[7] J. Lebovits, J. Lévy Véhel, and E. Herbin. Stochastic integration with respect to multifractional Brownian motion via tangent fractional Brownian motions. Stochastic Process. Appl., 124(1):678–708, 2014.

[8] D. Nualart and M.S. Taqqu. Wick-Itô formula for Gaussian processes. Stoch. Anal. Appl., 24(3):599–614, 2006.

Organised by: The T.N. Thiele Centre

Contact person: Søren Asmussen