Analysis of the Rosenblatt process

Abstract

Abstract. 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.Mathematics Subject Classification. 60G12, 60G15, 60H05, 60H07.

Bayesian estimation of self-similarity exponent

“…The Rosenblatt process is also known as Hermite process of the second order (note that the fractional Brownian motion is of the first order) [80,120]. Thus, in order to produce synthetic data for estimation purpose, I use here the wavelet-based synthesis of the Rosenblatt process as proposed and kindly provided by Abry and Pipiras [5].…”

From intersection local time to the Rosenblatt process

“…The double prime on the integral means that the diagonals λ 1 = ±λ 2 are excluded in the integration. The process also has a time representation as a double integral on R 2 with respect to Brownian motion, and a finite interval integral representation obtained in [39].…”

“…Significant attention has been attracted by the Rosenblatt process due to its mathematical interest, and possible applications where the Gaussian property may not be assumed. Recent papers on the subject include [39], which develops a related stochastic calculus and mentions areas of application (see also references therein), [23] and [40], where new properties of the process have been found, and [17], which provides a strong approximation for the process. Relevant information for the present paper on the Rosenblatt process and fractional Brownian motion is given in the next section.…”

“…The method of Taqqu, which was developed for Hermite processes generally [34], is based on limits of sums of strongly dependent random variables. The Rosenblatt process can also be defined as a double stochastic integral [36,39]. Our approach consists in deriving the Rosenblatt process from a specific random particle system, which hopefully provides an intuitive physical interpretation of this process.…”

Properties of trajectories of a multifractional Rosenblatt process

“…The most studied among these processes is the Hermite process of rank 2 defined in the paper by Rosenblatt [8] (the latter is also known as the Rosenblatt process). Among publications devoted to the Rosenblatt process, we mention papers by Pipiras [7], where a wavelet expansion is constructed for this process; Tudor [13], where stochastic analysis with respect to the Rosenblatt process is developed; Albin [1], where the distribution of the maximum of this process is found; and Tudor and Torres [12], where an application of the Rosenblatt process in finance mathematics is considered (namely, the Rosenblatt process is considered in [12] as a model of price evolutions).2010 Mathematics Subject Classification. Primary 60G22; Secondary 60J55, 60B10.…”