Analysis of the Rosenblatt process
نویسنده
چکیده
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 Major (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. 2000 AMS Classification Numbers: 60G12, 60G15, 60H05, 60H07
منابع مشابه
Asymptotic behavior of the Whittle estimator for the increments of a Rosenblatt process
The purpose of this paper is to estimate the self-similarity index of the Rosenblatt process by using the Whittle estimator. Via chaos expansion into multiple stochastic integrals, we establish a non-central limit theorem satisfied by this estimator. We illustrate our results by numerical simulations. 2000 AMS Classification Numbers: Primary: 60G18; Secondary 60F05, 60H05, 62F12.
متن کاملDonsker type theorem for the Rosenblatt process and a binary market model
In this paper, we prove a Donsker type approximation theorem for the Rosenblatt process, which is a selfsimilar stochastic process exhibiting long range dependence. By using numerical results and simulated data, we show that this approximation performs very well. We use this result to construct a binary market model driven by this process and we show that the model admits arbitrage opportunitie...
متن کاملWavelet-based synthesis of the Rosenblatt process
Based on a wavelet-type expansion of the Rosenblatt process, we introduce and examine two different practical ways to simulate the Rosenblatt process. The synthesis procedures proposed here are obtained by either truncating the series of the approximation term or using the approximation coefficients in the wavelet-type expansion of the Rosenblatt process. Both benefit from the low computational...
متن کاملA wavelet analysis of the Rosenblatt process: chaos expansion and estimation of the self-similarity parameter
The purpose of this paper is to make a wavelet analysis of self-similar stochastic processes by using the techniques of the Malliavin calculus and the chaos expansion into multiple stochastic integrals. Our examples are the fractional Brownian motion and the Rosenblatt process. We study the asymptotic behavior of the statistics based on the wavelet coefficients of these processes. We find that,...
متن کاملApproximating the Rosenblatt process by multiple Wiener integrals
Let Z be the Rosenblatt process with the representation Z t = ∫ t
متن کامل