Electronic Journal of Probability

Itô Formula and Local Time for the Fractional Brownian Sheet

Ciprian Tudor and Frederi Viens

Full-text: Open access

Abstract

Using the techniques of the stochastic calculus of variations for Gaussian processes, we derive an Itô formula for the fractional Brownian sheet with Hurst parameters bigger than $1/2$. As an application, we give a stochastic integral representation for the local time of the fractional Brownian sheet.

Article information

Source
Electron. J. Probab., Volume 8 (2003), paper no. 14, 31 p.

Dates
First available in Project Euclid: 23 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464037587

Digital Object Identifier
doi:10.1214/EJP.v8-155

Mathematical Reviews number (MathSciNet)
MR1998763

Zentralblatt MATH identifier
1067.60030

Subjects
Primary: 60H07: Stochastic calculus of variations and the Malliavin calculus
Secondary: 60G18: Self-similar processes 60G15: Gaussian processes 60J55: Local time and additive functionals

Keywords
fractional Brownian sheet Ito formula local time Tanaka formula Malliavin calculus

Citation

Tudor, Ciprian; Viens, Frederi. Itô Formula and Local Time for the Fractional Brownian Sheet. Electron. J. Probab. 8 (2003), paper no. 14, 31 p. doi:10.1214/EJP.v8-155. https://projecteuclid.org/euclid.ejp/1464037587


Export citation

References

  • E. Alòs, O. Mazet, D. Nualart (2001). Stochatic calculus with respect to Gaussian processes. Annals of probability, 29: 766-801.
  • E. Alòs, D. Nualart (2001). Stochastic integration with respect to the fractional Brownian motion. Preprint.
  • A. Ayache, S. Léger and M. Pontier (2002). Drap Brownien fractionnaire. Potential Analysis, 17(1), 31-43.
  • X. Bardina, M. Jolis and C.A. Tudor (2002). Weak convergence to the fractional Brownian sheet. Preprint núm. 06/2002, Universitat Autònoma de Barcelona.
  • S. Berman (1973). Local nondeterminism and local times of Gaussian processes. Indiana Univ. Math. J., 23: 69-94.
  • P. Carmona, L.Coutin (1998). Stochastic integration with respect to fractional Brownian motion. Preprint.
  • Cheridito, D. Nualart (2002). Stochastic integral of divergence type with respect to fBm with H \in (0;1/2). Preprint.
  • L. Coutin, D. Nualart and C.A. Tudor (2001). The Tanaka formula for the fractional Brownian motion. Stoc. Proc. Appl., 94(2):301-315.
  • L. Decreusefond, A. Ustunel (1998). Stochastic analysis of the fractional Brownian motion. Potential Analysis, 10:177-214.
  • M. Dozzi (1989). Stochastic processes with a multidimensional parameter. Longman Scientific and Technical.
  • T. E. Duncan, Y. Hu and B. Pasik-Duncan (2000). Stochastic calculus for fractional Brownian motion I. Theory. Siam J. Control Optim., 38(2):582-612.
  • M. Eddahbi, R. Lacayo, J.L. Sole, C.A. Tudor, J. Vives (2002). Regularity and asymptotic behaviour of the local time for the d-dimensional fractional Brownian motion with N-parameters. Preprint.
  • Y. Hu, B. Oksendal (2002). Chaos expansion of local time of fractional Brownian motions. Stoch. Analy. Appl., 20 (4): 815-837.
  • P. Imkeller (1984). Stochastic analysis and local time for (N.d)-Wiener process. Ann. Inst. Henri Poincaré, 20(1): 75-101.
  • B.B. Mandelbrot, J.W. Van Ness. Fractional Brownian motion, fractional noises and application. SIAM Review, 10(4):422-437.
  • D. Nualart. Une formule d'Itô pour les martingales continues à deux indices et quelques applications. Ann. Inst. Henri Poincaré, 20(3):251-275.
  • D. Nualart (1995). Malliavin Calculus and Related Topics. Springer V.
  • Y. Xiao, T. Zhang (2002). Local times of fractional Brownian sheets. Probab. Theory Relat. Fields, to appear.