Hostname: page-component-848d4c4894-mwx4w Total loading time: 0 Render date: 2024-06-20T10:51:30.933Z Has data issue: false hasContentIssue false
  • English
  • Français

Approximation of the fractional Brownian sheet VIA Ornstein-Uhlenbeck sheet

Published online by Cambridge University Press:  31 March 2007

Laure Coutin  and
Monique Pontier
Laure Coutin
Affiliation:
Université Paul Sabatier, 31062 Toulouse cedex 04; Laure.Coutin@lsp.ups-tlse.fr; Monique.Pontier@lsp.ups-tlse.fr
Monique Pontier
Affiliation:
Université Paul Sabatier, 31062 Toulouse cedex 04; Laure.Coutin@lsp.ups-tlse.fr; Monique.Pontier@lsp.ups-tlse.fr
Get access

Abstract

A stochastic “Fubini” lemma and an approximation theorem for integrals on the plane are used to produce a simulation algorithm for an anisotropic fractional Brownian sheet. The convergence rate is given. These results are valuable for any value of the Hurst parameters $(\alpha_1,\alpha_2)\in ]0,1[^2,\alpha_i\neq\frac{1}{2}.$ Finally, the approximation process is iterative on the quarter plane $\mathbb {R}_+^2.$ A sample of such simulations can be used to test estimators of the parameters αi,i = 1,2.

Keywords

random field simulation and approximationanisotropic fractional Brownian sheet.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 11: Special Issue: "Stochastic analysis and mathematical finance" in honor of Nicole El Karoui's 60th birthdaySpecial Issue: "Stochastic analysis and mathematical finance" in honor of Nicole El Karoui's 60th birthday , June 2007 , pp. 115 - 146
Copyright
© EDP Sciences, SMAI, 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

J. Audounet, G. Montseny and B. Mbodje, A simple viscoelastic damper model — application to a vibrating string. Analysis and optimization of systems: state and frequency domain approaches for infinite-dimensional systems (Sophia-Antipolis, 1992), Lect. Notes Control Inform. Sci. 185, Springer, Berlin (1993) 436–446.
Ayache, A., Léger, S. and Pontier, M., Les ondelettes à la conquête du drap brownien fractionnaire. CRAS série I 335 (2002) 1063–1068.
Ayache, A. and Taqqu, M., Rate optimality of wavelet series approximations of fractional Brownian motion. J. Fourier Anal. Appl. 9 (2003) 451–471. CrossRef
J.M. Bardet, G. Lang, G. Oppenheim, A. Philippe and M. Taqqu, Generators of long-range dependent processes: a survey, in Long-Range dependence, Theory and Applications. Birkhauser (2003) 579–623.
O.E. Barndorff-Nielsen and N. Shephard, Non Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics. J.R. Statistical Society B 63 (2001) 167–241.
Bernam, S., Gaussian processes with stationary increments local times and sample function properties. Ann. Math. Statist. 41 (1970) 1260–1272.
Carmona, P., Coutin, L. and Montseny, G., Approximation of some Gaussian processes. Stat. Inference of Stoch. Processes 3 (2000) 161–171. CrossRef
S. Cohen, Champs localement auto-similaires, dans Lois d'échelle, fractales et ondelettes 1, P. Abry, P. Goncalvès, J. Lévy Véhel, Eds. (2001).
X.M. Fernique, Régularité des trajectoires des fonctions aléatoires gaussiennes, in École d'été de probabilités de saint-Flour L. N. in Math 480 (1974) 1–96.
Igloi, E. and Terdik, G., Long-range dependence through gamma-mixed Ornstein-Uhlenbeck process. E.J.P. 4 (1999) 1–33.
N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes. North-Holland, Amsterdam (1981).
I. Karatzas and S.E. Schreve, Brownian Motion and Stochastic Calculus. Springer, 2d edition (1999).
S. Léger, Drap brownien fractionnaire, thèse à l'Université d'Orléans (2000).
S. Léger and M. Pontier, Drap brownien fractionnaire, in C.R.A.S., Paris, série I 329 (1999) 893–898.
Meyer, Y., Sellan, F. and Taqqu, M., Wavelets, generalized white noise and fractional integration: the synthesis of fractional Brownian motion. Journal of Fourier Analysis and Applications 5 (1999) 465–494. CrossRef
D. Revuz and M. Yor, Continuous Martingales and Brownian Motion. Springer-Verlag, Berlin (1990).
G. Samorodnitsky and M. Taqqu, Stable Non-Gaussian random Processes, Stochastic Modeling. Chapman and Hall, New York (1994).
D.W. Stroock, A Concise Introduction to the Theory of Integration Stochastic Integration. Birkhauser, 2d edition (1994).
Wood, A.T.A. and Chan, G., Simulation, A of stationary Gaussian processes in [0,1]d. J. Comput. Graphical Statist. 3–4 (1994) 409–432.