No abstract

The statistical analysis for equations driven by fractional Gaussian process (fGp) is relatively recent. The development of stochastic calculus with respect to the fGp allowed to study such models. In the present paper we consider the drift parameter estimation problem for the non-ergodic Ornstein-Uhlenbeck process defined as dX t = θX t dt + dG t , t ≥ 0 with an unknown parameter θ > 0, where G is a Gaussian process. We provide sufficient conditions, based on the properties of G, ensuring the strong consistency and the asymptotic distribution of our estimator θ t of θ based on the observation {X s , s ∈ [0, t]} as t → ∞. Our approach offers an elementary, unifying proof of [4], and it allows to extend the result of [4] to the case when G is a fractional Brownian motion with Hurst parameter H ∈ (0, 1). We also discuss the cases of subfractional Brownian motion and bifractional Brownian motion.

Using the Malliavin calculus with respect to Gaussian processes and the multiple stochastic integrals we derive Itô's and Tanaka's formulas for the d-dimensional bifractional Brownian motion.2000 AMS Classification Numbers: 60G12, 60G15, 60H05, 60H07.

In this paper we introduce and study a self-similar Gaussian process that is the bifractional Brownian motion B H,K with parameters H ∈ (0, 1) and K ∈ (1, 2) such that HK ∈ (0, 1). A remarkable difference between the case K ∈ (0, 1) and our situation is that this process is a semimartingale when 2HK = 1.2000 Mathematics Subject Classification. Primary 60G15; Secondary 60G18.

We first study the drift parameter estimation of the fractional Ornstein-Uhlenbeck process (fOU) with periodic mean for every 1 2 < H < 1. More precisely, we extend the consistency proved in [6] for 1 2 < H < 3 4 to the strong consistency for any 1 2 < H < 1 on the one hand, and on the other, we also discuss the asymptotic normality given in [6]. In the second main part of the paper, we study the strong consistency and the asymptotic normality of the fOU of the second kind with periodic mean for any 1 2 < H < 1.