Analysis of the Rosenblatt process

Tudor, Ciprian A.

ESAIM: Probability and Statistics, Tome 12 (2008), pp. 230-257.

We analyze the Rosenblatt process which is a selfsimilar process with stationary increments and which appears as limit in the so-called Non Central Limit Theorem (Dobrushin and Majòr (1979), Taqqu (1979)). This process is non-gaussian and it lives in the second Wiener chaos. We give its representation as a Wiener-Itô multiple integral with respect to the brownian motion on a finite interval and we develop a stochastic calculus with respect to it by using both pathwise type calculus and Malliavin calculus.

[1] P. Abry and V. Pipiras, Wavelet-based synthesis of the Rosenblatt process. Signal Process. 86 (2006) 2326-2339. | Zbl

[2] J.M.P. Albin, A note on the Rosenblatt distributions. Statist. Probab. Lett. 40 (1998) 83-91. | MR | Zbl

[3] J.M.P. Albin, On extremal theory for self similar processes. Ann. Probab. 26 (1998) 743-793. | MR | Zbl

[4] E. Alòs, O. Mazet and D. Nualart, Stochastic calculus with respect to Gaussian processes. Ann. Probab. 29 (2001) 766-801. | MR | Zbl

[5] E. Alòs and D. Nualart, Stochastic integration with respect to the fractional Brownian motion. Stoch. Stoch. Rep. 75 (2003) 129-152. | MR | Zbl

[6] T. Androshuk and Y. Mishura, Mixed Brownian-fractional Brownian model: absence of arbitrage and related topics. Stochastics An Int. J. Probability Stochastic Processes 78 (2006) 281-300. | MR | Zbl

[7] F. Biagini, M. Campanino and S. Fuschini, Discrete approximation of stochastic integrals with respect of fractional Brownian motion of Hurst index $H>1/2.$ Preprint University of Bologna (2005). | MR | Zbl

[8] P. Cheridito, H. Kawaguchi and M. Maejima, Fractional Ornstein-Uhlenbeck processes. Electron. J. Probab. 8 (2003) 1-14. | EuDML | MR | Zbl

[9] L. Decreusefond and A.S. Ustunel, Stochastic analysis of the fractional Brownian motion. Potential Anal. 10 (1998) 177-214. | MR | Zbl

[10] G. Da Prato and J. Zabczyk, Stochastic equations in infinite dimensions. Cambridge University Press (1992). | MR | Zbl

[11] R.L. Dobrushin and P. Major, Non-central limit theorems for non-linear functionals of Gaussian fields. Z. Wahrscheinlichkeitstheorie verw. Gebiete 50 (1979) 27-52. | MR | Zbl

[12] A. Drewitz, Mild solutions to stochastic evolution equations with fractional Brownian motion. Diploma thesis at TU Darmstadt (2005).

[13] P. Embrechts and M. Maejima, Selfsimilar processes. Princeton University Press, Princeton, New York (2002). | MR | Zbl

[14] R. Fox and M.S. Taqqu, Multiple stochastic integrals with dependent integrators. J. Mult. Anal. 21 (1987) 105-127. | MR | Zbl

[15] V. Goodman and J. Kuelbs, Gaussian chaos and functional law of the ierated logarithm for Itô-Wiener integrals. Ann. I.H.P., Section B 29 (1993) 485-512. | Numdam | MR | Zbl

[16] M. Gradinaru, I. Nourdin, F. Russo and P. Vallois, $m$-order integrals and generalized Itôs formula; the case of a fractional Brownian motion with any Hurst parameter. Preprint, to appear in Annales de l'Institut Henri Poincaré (2003). | Numdam | Zbl

[17] M. Gradinaru, I. Nourdin and S. Tindel, Ito's and Tanaka's type formulae for the stochastic heat equation. J. Funct. Anal. 228 (2005) 114-143. | MR | Zbl

[18] P. Hall, W. Hardle, T. Kleinow and P. Schmidt, Semiparametric Bootstrap Approach to Hypothesis tests and Confidence intervals for the Hurst coefficient. Stat. Infer. Stoch. Process. 3 (2000) 263-276. | MR | Zbl

[19] M. Jolis and M. Sanz, Integrator properties of the Skorohod integral. Stochastics and Stochastics Reports 41 (1992) 163-176. | MR | Zbl

[20] O. Kallenberg, On an independence criterion for multiple Wiener integrals. Ann. Probab. 19 (1991) 483-485. | MR | Zbl

[21] H. Kettani and J. Gubner, Estimation of the long-range dependence parameter of fractional Brownian motionin, in Proc. 28th IEEE LCN03 (2003).

[22] I. Kruk, F. Russo and C.A. Tudor, Wiener integrals, Malliavin calculus and covariance measure structure. J. Funct. Anal. 249 (2007) 92-142. | MR | Zbl

[23] N.N. Leonenko and V.V. Ahn, Rate of convergence to the Rosenblatt distribution for additive functionals of stochastic processes with long-range dependence. J. Appl. Math. Stoch. Anal. 14 (2001) 27-46. | MR | Zbl

[24] N.N. Leonenko and W. Woyczynski, Scaling limits of solutions of the heat equation for singular Non-Gaussian data. J. Stat. Phys. 91 423-438. | MR | Zbl

[25] M. Maejima and C.A. Tudor, Wiener integrals with respect to the Hermite process and a non central limit theorem. Stoch. Anal. Appl. 25 (2007) 1043-1056. | MR | Zbl

[26] O. Mocioalca and F. Viens, Skorohod integration and stochastic calculus beyond the fractional Brownian scale. J. Funct. Anal. 222 (2004) 385-434. | MR | Zbl

[27] I. Norros, E. Valkeila and J. Virtamo, An elementary approach to a Girsanov formula and other analytical results for fractional Brownian motion. Bernoulli 5 (1999) 571-587. | MR | Zbl

[28] I. Nourdin, A simple theory for the study of SDEs driven by a fractional Brownian motion, in dimension one. Séminaire de Probabilités XLI (2006). To appear. | MR | Zbl

[29] D. Nualart, Malliavin Calculus and Related Topics. Springer (1995). | MR | Zbl

[30] D. Nualart and M. Zakai, Generalized mulptiple stochastic integrals and the representation of Wiener functionals. Stochastics 23 (1987) 311-330. | MR | Zbl

[31] V. Pipiras, Wavelet type expansion of the Rosenblatt process. J. Fourier Anal. Appl. 10 (2004) 599-634. | MR | Zbl

[32] V. Pipiras and M. Taqqu, Convergence of weighted sums of random variables with long range dependence. Stoch. Process. Appl. 90 (2000) 157-174. | MR | Zbl

[33] V. Pipiras and Murad Taqqu, Integration questions related to the fractional Brownian motion. Probab. Theor. Relat. Fields 118 (2001) 251-281. | MR | Zbl

[34] N. Privault and C.A. Tudor, Skorohod and pathwise stochastic calculus with respect to an $L^2$-process. Rand. Oper. Stoch. Equ. 8 (2000) 201-204. | Zbl

[35] Z. Qian and T. Lyons, System control and rough paths. Clarendon Press, Oxford (2002). | MR | Zbl

[36] M. Rosenblatt, Independence and dependence. Proc. 4th Berkeley Symposium on Math, Stat. II (1961) 431-443. | MR | Zbl

[37] F. Russo and P. Vallois, Forward backward and symmetric stochastic integration. Probab. Theor. Relat. Fields 97 (1993) 403-421. | MR | Zbl

[38] F. Russo and P. Vallois, Stochastic calculus with respect to a finite quadratic variation process. Stoch. Stoch. Rep. 70 (2000) 1-40. | MR | Zbl

[39] F. Russo and P. Vallois, Elements of stochastic calculus via regularization. Preprint, to appear in Séminaire de Probabilités (2006). | MR | Zbl

[40] G. Samorodnitsky and M. Taqqu, Stable Non-Gaussian random variables. Chapman and Hall, London (1994). | MR

[41] A.S. Üstunel and M. Zakai, On independence and conditioning on Wiener space. Ann. Probab. 17 (1989) 1441-1453. | MR | Zbl

[42] M. Taqqu, Weak convergence to the fractional Brownian motion and to the Rosenblatt process. Z. Wahrscheinlichkeitstheorie verw. Gebiete 31 (1975) 287-302. | MR | Zbl

[43] M. Taqqu, Convergence of integrated processes of arbitrary Hermite rank. Z. Wahrscheinlichkeitstheorie verw. Gebiete 50 (1979) 53-83. | MR | Zbl

[44] M. Taqqu, A bibliographical guide to selfsimilar processes and long-range dependence. Dependence in Probability and Statistics, Birkhauser, Boston (1986) 137-162. | MR | Zbl

[45] S. Tindel, C.A. Tudor and F. Viens, Stochastic evolution equations with fractional Brownian motion. Probab. Theor. Relat. Fields. 127 (2003) 186-204. | MR | Zbl

[46] C.A. Tudor, Itô's formula for the infinite-dimensional fractional Brownian motion. J. Math. Kyoto University 45 (2005) 531-546. | MR | Zbl

[47] W.B. Wu, Unit root testing for functionals of linear processes. Econ. Theory 22 (2005) 1-14. | MR | Zbl