Approximation of the fractional brownian sheet via Ornstein-Uhlenbeck sheet

Coutin, Laure; Pontier, Monique

ESAIM: Probability and Statistics, Tome 11 (2007), pp. 115-146.

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 $({\alpha }_1,{\alpha }_2)\in {]0,1[}^2,{\alpha }_i\ne \frac{1}{2}.$ Finally, the approximation process is iterative on the quarter plane ${ℝ}_{+}^2.$ A sample of such simulations can be used to test estimators of the parameters ${\alpha }_i,i=1,2.$

[1] 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. | Zbl

[2] A. Ayache, S. Léger and M. Pontier, Les ondelettes à la conquête du drap brownien fractionnaire. CRAS série I 335 (2002) 1063-1068. | Zbl

[3] A. Ayache and M. Taqqu, Rate optimality of wavelet series approximations of fractional Brownian motion. J. Fourier Anal. Appl. 9 (2003) 451-471. | Zbl

[4] 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. | Zbl

[5] 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. | Zbl

[6] S. Bernam, Gaussian processes with stationary increments local times and sample function properties. Ann. Math. Statist. 41 (1970) 1260-1272. | Zbl

[7] P. Carmona, L. Coutin and G. Montseny, Approximation of some Gaussian processes. Stat. Inference of Stoch. Processes 3 (2000) 161-171. | Zbl

[8] S. Cohen, Champs localement auto-similaires, dans Lois d'échelle, fractales et ondelettes 1, P. Abry, P. Goncalvès, J. Lévy Véhel, Eds. (2001).

[9] 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. | Zbl

[10] E. Igloi and G. Terdik, Long-range dependence through gamma-mixed Ornstein-Uhlenbeck process. E.J.P. 4 (1999) 1-33. | Zbl

[11] N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes. North-Holland, Amsterdam (1981). | MR | Zbl

[12] I. Karatzas and S.E. Schreve, Brownian Motion and Stochastic Calculus. Springer, 2d edition (1999). | Zbl

[13] S. Léger, Drap brownien fractionnaire, thèse à l'Université d'Orléans (2000).

[14] S. Léger and M. Pontier, Drap brownien fractionnaire, in C.R.A.S., Paris, série I 329 (1999) 893-898. | Zbl

[15] Y. Meyer, F. Sellan and M. Taqqu, Wavelets, generalized white noise and fractional integration: the synthesis of fractional Brownian motion. Journal of Fourier Analysis and Applications 5 (1999) 465-494. | Zbl

[16] D. Revuz and M. Yor, Continuous Martingales and Brownian Motion. Springer-Verlag, Berlin (1990). | Zbl

[17] G. Samorodnitsky and M. Taqqu, Stable Non-Gaussian random Processes, Stochastic Modeling. Chapman and Hall, New York (1994). | MR | Zbl

[18] D.W. Stroock, A Concise Introduction to the Theory of Integration Stochastic Integration. Birkhauser, 2d edition (1994). | MR | Zbl

[19] A.T.A. Wood and G. Chan, A Simulation of stationary Gaussian processes in ${[0,1]}^d$. J. Comput. Graphical Statist. 3-4 (1994) 409-432.