Ehsan Azmoodeh

Drift parameter estimation for fractional Ornstein–Uhlenbeck process of the second kind

The fractional Ornstein–Uhlenbeck process of the second kind (fOU2) is the solution of the Langevin equation with driving noise where B is a fractional Brownian motion with Hurst parameter H∈(0, 1). In this article, in the case H>½, we prove that the least-squares estimator introduced in [Hu Y, Nualart D. Parameter estimation for fractional Ornstein–Uhlenbeck processes. Stat. Probab. Lett. 2010;80(11–12):1030–1038], provides a consistent estimator. Moreover, using central limit theorem for multiple Wiener integrals, we prove asymptotic normality of the estimator valid for the whole range H∈(½, 1).

Generalization of the Nualart-Peccati criterion

The celebrated Nualart–Peccati criterion [Ann. Probab. 33 (2005) 177–193] ensures the convergence in distribution toward a standard Gaussian random variable N of a given sequence {Xn}n≥1 of multiple Wiener–Ito integrals of fixed order, if E[X2n]→1 and E[X4n]→E[N4]=3. Since its appearance in 2005, the natural question of ascertaining which other moments can replace the fourth moment in the above criterion has remained entirely open. Based on the technique recently introduced in [J. Funct. Anal. 266 (2014) 2341–2359], we settle this problem and establish that the convergence of any even moment, greater than four, to the corresponding moment of the standard Gaussian distribution, guarantees the central convergence. As a by-product, we provide many new moment inequalities for multiple Wiener–Ito integrals. For instance, if X is a normalized multiple Wiener–Ito integral of order greater than one, ∀k≥2,E[X2k]>E[N2k]=(2k−1)!!.

Convergence Towards Linear Combinations of Chi-Squared Random Variables: A Malliavin-Based Approach

We investigate the problem of finding necessary and sufficient conditions for convergence in distribution towards a general finite linear combination of independent chi-squared random variables, within the framework of random objects living on a fixed Gaussian space. Using a recent representation of cumulants in terms of the Malliavin calculus operators \(\Gamma _{i}\) (introduced by Nourdin and Peccati, J. Appl. Funct. Anal. 258(11), 3775–3791, 2010), we provide conditions that apply to random variables living in a finite sum of Wiener chaoses. As an important by-product of our analysis, we shall derive a new proof and a new interpretation of a recent finding by Nourdin and Poly (Electron. Commun. Probab. 17(36), 1–12, 2012), concerning the limiting behavior of random variables living in a Wiener chaos of order two. Our analysis contributes to a fertile line of research, that originates from questions raised by Marc Yor, in the framework of limit theorems for non-linear functionals of Brownian local times.

Rate of Convergence for Discretization of Integrals with Respect to Fractional Brownian Motion

In this article, a uniform discretization of stochastic integrals

Asymptotic normality of randomized periodogram for estimating quadratic variation in mixed Brownian–fractional Brownian model

\int _{0}^{1} f^{\prime }_-(B_t)\mathrm d B_t

Parameter estimation based on discrete observations of fractional Ornstein–Uhlenbeck process of the second kind

∫01f−′(Bt)dBt, where

Necessary and sufficient conditions for Hölder continuity of Gaussian processes

B