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

Fractional Ornstein-Uhlenbeck process of the second kind $(\text{fOU}_{2})$ is solution of the Langevin equation $\mathrm{d}X_t = -\theta X_t\,\mathrm{d}t+\mathrm{d}Y_t^{(1)}, \ \theta >0$ with Gaussian driving noise $Y_t^{(1)} := \int^t_0 e^{-s} \,\mathrm{d}B_{a_s}$, where $a_t= H e^{\frac{t}{H}}$ and $B$ is a fractional Brownian motion with Hurst parameter $H \in (0,1)$. In this article, we consider the case $H>\frac{1}{2}$. Then using the ergodicity of $\text{fOU}_{2}$ process, we construct consistent estimators of drift parameter $\theta$ based on discrete observations in two possible cases: $(i)$ the Hurst parameter $H$ is known and $(ii)$ the Hurst parameter $H$ is unknown. Moreover, using Malliavin calculus technique, we prove central limit theorems for our estimators which is valid for the whole range $H \in (\frac{1}{2},1)$.Comment: Modified version. arXiv admin note: text overlap with arXiv:1302.604

A general approach to small deviation via concentration of measures

We provide a general approach to obtain upper bounds for small deviations $\mathbb{P}(\Vert y \Vert \le \epsilon)$ in different norms, namely the supremum and $\beta$- H\"older norms. The large class of processes $y$ under consideration takes the form $y_t= X_t + \int_0^t a_s d s$, where $X$ and $a$ are two possibly dependent stochastic processes. Our approach provides an upper bound for small deviations whenever upper bounds for the \textit{concentration of measures} of $L^p$- norm of random vectors built from increments of the process $X$ and \textit{large deviation} estimates for the process $a$ are available. Using our method, among others, we obtain the optimal rates of small deviations in supremum and $\beta$- H\"older norms for fractional Brownian motion with Hurst parameter $H\le\ \frac{1}{2}$. As an application, we discuss the usefulness of our upper bounds for small deviations in pathwise stochastic integral representation of random variables motivated by the hedging problem in mathematical finance

Optimal Berry-Esseen bounds on the Poisson space

We establish new lower bounds for the normal approximation in the Wasserstein distance of random variables that are functionals of a Poisson measure. Our results generalize previous findings by Nourdin and Peccati (2012, 2015) and Bierm\'e, Bonami, Nourdin and Peccati (2013), involving random variables living on a Gaussian space. Applications are given to optimal Berry-Esseen bounds for edge counting in random geometric graphs

Fourth Moment Theorems for Markov Diffusion Generators

Inspired by the insightful article arXiv:1210.7587, we revisit the Nualart-Peccati-criterion arXiv:math/0503598 (now known as the Fourth Moment Theorem) from the point of view of spectral theory of general Markov diffusion generators. We are not only able to drastically simplify all of its previous proofs, but also to provide new settings of diffusive generators (Laguerre, Jacobi) where such a criterion holds. Convergence towards gamma and beta distributions under moment conditions is also discussed.Comment: 15 page