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We propose a modified least squares estimator for the drift parameter of the Ornstein–Uhlenbeck process when the observations are available at a discrete instant in a low-frequency level. Unlike in the past literature, this modified least squares estimator is asymptotically unbiased. This estimator is combined with the ergodic theorem to obtain joint estimators ( θ ˆ n , σ ˆ n 2) for both drift and diffusion parameters (θ , σ 2). For any fixed observation time step h > 0 , the strong consistency and joint asymptotic normality of our estimators ( θ ˆ n , σ ˆ n 2) are obtained by using the linear model technology and the Delta method. Surprisingly, this linear model technology is not new but it is used for the first time to our model. It very significantly simplifies the arguments that researchers used in the literature. Numerical results illustrate the asymptotic behavior of the estimators. [ABSTRACT FROM AUTHOR]

The goal of this paper is to construct ergodic estimators for double exponential Ornstein–Uhlenbeck process, where the process is observed at discrete time instants with time step size h. We show the existence and uniqueness of the function equations to determine the estimators for fixed time step size h. Also, we show the strong consistency and the asymptotic normality of the estimators. Furthermore, we propose a simulation method of the double exponential Ornstein–Uhlenbeck process and perform some numerical simulations to demonstrate the effectiveness of the proposed estimators. [ABSTRACT FROM AUTHOR]

Assuming that a threshold Ornstein–Uhlenbeck process is observed at discrete time instants, we propose generalized moment estimators to estimate the parameters. Our theoretical basis is the celebrated ergodic theorem. With the sampling time step arbitrarily fixed, we prove the strong consistency and asymptotic normality of our estimators as the sample size tends to infinity. [ABSTRACT FROM AUTHOR]

Abstract: The aim of this paper is to establish the uniform convergence of the densities of a sequence of random variables, which are functionals of an underlying Gaussian process, to a normal density. Precise estimates for the uniform distance are derived by using the techniques of Malliavin calculus, combined with Steinʼs method for normal approximation. We need to assume some non-degeneracy conditions. First, the study is focused on random variables in a fixed Wiener chaos, and later, the results are extended to the uniform convergence of the derivatives of the densities and to the case of random vectors in some fixed chaos, which are uniformly non-degenerate in the sense of Malliavin calculus. Explicit upper bounds for the uniform norm are obtained for random variables in the second Wiener chaos, and an application to the convergence of densities of the least square estimator for the drift parameter in Ornstein–Uhlenbeck processes is discussed. [Copyright &y& Elsevier]

Abstract: We study the problem of parameter estimation for generalized Ornstein–Uhlenbeck processes driven by -stable noises, observed at discrete time instants. Least squares method is used to obtain an asymptotically consistent estimator. The strong consistency and the rate of convergence of the estimator have been studied. The estimator has a higher order of convergence in the general stable, non-Gaussian case than in the classical Gaussian case. [Copyright &y& Elsevier]