Parameter estimation for Gaussian mean-reverting Ornstein–Uhlenbeck processes of the second kind: Non-ergodic case

We consider a least square-type method to estimate the drift parameters for the mean-reverting Ornstein–Uhlenbeck process of the second kind [Formula: see text] defined as [Formula: see text], with unknown parameters [Formula: see text] and [Formula: see text], where [Formula: see text] with [Formula: see text], and [Formula: see text] is a Gaussian process. In order to establish the consistency and the asymptotic distribution of least square-type estimators of [Formula: see text] and [Formula: see text] based on the continuous-time observations [Formula: see text] as [Formula: see text], we impose some technical conditions on the process [Formula: see text], which are satisfied, for instance, if [Formula: see text] is a fractional Brownian motion with Hurst parameter [Formula: see text], [Formula: see text] is a subfractional Brownian motion with Hurst parameter [Formula: see text] or [Formula: see text] is a bifractional Brownian motion with Hurst parameters [Formula: see text]. Our method is based on pathwise properties of [Formula: see text] and [Formula: see text] proved in the sequel.

The Structure of Autocovariance Matrix of Discrete Time Subfractional Brownian Motion

This article explores the structure of autocovariance matrix of discrete time subfractional Brownian motion and obtains an approximation theorem and a structure theorem to the autocovariance matrix of this stochastic process. Moreover, we give an expression to the unique time varying eigenvalue of the autocovariance matrix in asymptotic means and prove that the increments of subfractional Brownian motion are asymptotic stationary processes. At last, we illustrate these results with numerical experiments and give some probable applications in finite impulse response filter.

On limit theorems of some extensions of fractional Brownian motion and their additive functionals

This paper is divided into two parts. The first deals with some limit theorems to certain extensions of fractional Brownian motion like: bifractional Brownian motion, subfractional Brownian motion and weighted fractional Brownian motion. In the second part we give the similar results of their continuous additive functionals; more precisely, local time and its fractional derivatives involving slowly varying function.