Electronic Communications in Probability

A note on ergodic transformations of self-similar Volterra Gaussian processes

Céline Jost

Full-text: Open access

Abstract

We derive a class of ergodic transformation of self-similar Gaussian processes that are Volterra, i.e. of type $X_t = \int^t_0 z_X(t,s)dW_s$, $t \in [0,\infty)$, where $z_X$ is a deterministic kernel and $W$ is a standard Brownian motion.

Article information

Source
Electron. Commun. Probab., Volume 12 (2007), paper no. 25, 259-266.

Dates
Accepted: 25 August 2007
First available in Project Euclid: 6 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465224968

Digital Object Identifier
doi:10.1214/ECP.v12-1298

Mathematical Reviews number (MathSciNet)
MR2335896

Zentralblatt MATH identifier
1129.60037

Subjects
Primary: 60G15: Gaussian processes
Secondary: 60G18: Self-similar processes 37A25: Ergodicity, mixing, rates of mixing

Keywords
Volterra Gaussian process Self-similar process Ergodic transformation Fractional Brownian motion

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Jost, Céline. A note on ergodic transformations of self-similar Volterra Gaussian processes. Electron. Commun. Probab. 12 (2007), paper no. 25, 259--266. doi:10.1214/ECP.v12-1298. https://projecteuclid.org/euclid.ecp/1465224968


Export citation

References

  • P. Deheuvels. Invariance of Wiener processes and of Brownian bridges by integral transforms and applications. Stochastic Processes and their Applications 13, 311-318, 1982.
  • P. Embrechts, M. Maejima. Selfsimilar Processes. Princeton University Press, 2002.
  • A. Erdélyi, W. Magnus, F. Oberhettinger, F.G. Tricomi. Tables of integral transforms, Volume 1. McGraw-Hill Book Company, Inc., 1954.
  • T. Hida, M. Hitsuda. Gaussian Processes. Translations of Mathematical Monographs 120, American Mathematical Society, 1993.
  • S.T. Huang, S. Cambanis. Stochastic and Multiple Wiener Integrals for Gaussian Processes. Annals of Probability 6(4), 585-614, 1978.
  • T. Jeulin, M. Yor. Filtration des ponts browniens et équations différentielles stochastiques linéaires. Séminaire de probabilités de Strasbourg 24, 227-265, 1990.
  • C. Jost. Measure-preserving transformations of Volterra Gaussian processes and related bridges. [arXiv:math.PR/0701888]
  • G.M. Molchan. Linear problems for a fractional Brownian motion: group approach. Theory of Probability and its Applications 47 (1), 69-78, 2003.
  • I. Norros, E. Valkeila, J. Virtamo. An elementary approach to a Girsanov formula and other analytical results on fractional Brownian motions. Bernoulli 5(4), 571-587, 1999.
  • G. Peccati. Explicit formulae for time-space Brownian chaos. Bernoulli 9(1), 25-48, 2003.
  • K. Petersen. Ergodic theory. Cambridge University Press, 1983.
  • A.N. Shiryaev. Probability, Second Edition. Springer, 1989.
  • A.M. Yaglom. Correlation Theory of Stationary and Related Random Functions, Volume 1: Basic Results. Springer, 1987.