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2014
We investigate the asymptotic behavior of the maximum likelihood estimators of the unknown parameters of positive recurrent Ornstein-Uhlenbeck processes driven by Ornstein-Uhlenbeck processes.
2016
It is considered Ornstein-Uhlenbeck process $ x_t = x_0 e^{-\\theta t} + \\mu (1-e^{-\\theta t}) + \\sigma \\int_0^t e^{-\\theta (t-s)} dW_s$, where $x_0 \\in R$, $\\theta>0$, $ \\mu \\in R$ and $\\sigma > 0$ are parameters. By use values $(z_k)_{k \\in N}$ of corresponding trajectories at a fixed positive moment $t$, a consistent estimate of each unknown parameter of the Ornstein-Uhlenbeck's stochastic process is constructed under assumption that all another parameters are known.
2012
In this paper, we investigate a sequential maximum likelihood estimator of the unknown drift parameter for a class of reflected generalized Ornstein-Uhlenbeck processes driven by spectrally positive Lévy processes. In both of the cases of negative drift and positive drift, we prove that the sequential maximum likelihood estimator of the drift parameter is closed, unbiased, normally distributed and strongly consistent. Finally a numerical test is presented to illustrate the efficiency of the estimator.
2021
In this work, we study the class of stochastic process that generalizes the OrnsteinUhlenbeck processes, hereafter called by Generalized Ornstein-Uhlenbeck Type Process and denoted by GOU type process. We consider them driven by the class of noise processes such as Brownian motion, symmetric α-stable Lévy process, a Lévy process, and even a Poisson process. We give necessary and sufficient conditions under the memory kernel function for the time-stationary and the Markov properties for these processes. When the GOU type process is driven by a Lévy noise we prove that it is infinitely divisible showing its generating triplet. Several examples derived from the GOU type process are illustrated showing some of their basic properties as well as some time series realizations. These examples also present their theoretical and empirical autocorrelation or normalized codifference functions depending on whether the process has a finite or infinite second moment. We also present the maximum li...
We consider the parameter estimation problem for the Ornstein-Uhlenbeck process X driven by a fractional Ornstein-Uhlenbeck process V , i.e. the pair of processes defined by the non-Markovian continuous-time long-memory dynamics dX t = −θX t dt + dV t ; t 0, with dV t = −ρV t dt + dB H t ; t 0, where θ > 0 and ρ > 0 are unknown parameters, and B H is a fractional Brownian motion of Hurst index H ∈ ( 1 2 , 1). We study the strong consistency as well as the asymptotic normality of the joint least squares estimator θ T , ρ T of the pair (θ, ρ), based either on continuous or discrete observations of {X s ; s ∈ [0, T ]} as the horizon T increases to +∞. Both cases qualify formally as partial-hbobservation questions since V is unobserved. In the latter case, several discretization options are considered. Our proofs of asymptotic normality based on discrete data, rely on increasingly strict restrictions on the sampling frequency as one reduces the extent of sources of observation. The strategy for proving the asymptotic properties is to study the case of continuous-time observations using the Malliavin calculus, and then to exploit the fact that each discrete-data estimator can be considered as a perturbation of the continuous one in a mathematically precise way, despite the fact that the implementation of the discrete-time estimators is distant from the continuous estimator. In this sense, we contend that the continuous-time estimator cannot be implemented in practice in any naïve way, and serves only as a mathematical tool in the study of the discrete-time estimators' asymptotics.
Annals of the Institute of Statistical Mathematics, 2012
Consider non-recurrent Ornstein-Uhlenbeck processes with unknown drift and diffusion parameters. Our purpose is to estimate the parameters jointly from discrete observations with a certain asymptotics. We show that the likelihood ratio of the discrete samples has the uniform LAMN property, and that some kind of approximated MLE is asymptotically optimal in a sense of asymptotic maximum concentration probability. The estimator is also asymptotically efficient in ergodic cases.
Theory of Probability and Mathematical Statistics, 2013
We consider parameter estimation for a process of Ornstein-Uhlenbeck type with reciprocal gamma marginal distribution, to be called reciprocal gamma Ornstein-Uhlenbeck (RGOU) process. We derive minimum contrast estimators of unknown parameters based on both the discrete and the continuous observations from the process as well as moments based estimators based on discrete observations. We prove that proposed estimators are consistent and asymptotically normal. The explicit forms of the asymptotic covariance matrices are determined by using the higher order spectral densities and cumulants of the RGOU process.
Scandinavian Journal of Statistics, 2021
Generalizations of the Ornstein-Uhlenbeck process defined through Langevin equations, such as fractional Ornstein-Uhlenbeck processes, have recently received a lot of attention. However, most of the literature focuses on the one-dimensional case with Gaussian noise. In particular, estimation of the unknown parameter is widely studied under Gaussian stationary increment noise. In this article, we consider estimation of the unknown model parameter in the multidimensional version of the Langevin equation, where the parameter is a matrix and the noise is a general, not necessarily Gaussian, vector-valued process with stationary increments. Based on algebraic Riccati equations, we construct an estimator for the parameter matrix. Moreover, we prove the consistency of the estimator and derive its limiting distribution under natural assumptions. In addition, to motivate our work, we prove that the Langevin equation characterizes essentially all multidimensional stationary processes.
Advances in Applied Probability, 2016
In this paper we consider an Ornstein-Uhlenbeck ( ) process (M (t)) t 0 whose parameters are determined by an external Markov process (X(t)) t 0 on a nite state space {1, . . . , d}; this process is usually referred to as Markov-modulated Ornstein-Uhlenbeck (or:
Stochastics
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2021
This paper presents a nonlinear autoregressive model by Ornstein Uhlenbeck processes innovation driven with white noise. The presented notations and preliminaries about these processes, have important applications in finance. The parameter estimation for these processes is constructed from the time-continuous likelihood function that leads to an explicit maximum likelihood estimator. A semiparametric method is proposed to estimate the nonlinear autoregressive function using the conditional least square method for parametric estimation, and a nonparametric kernel approach by using the nonparametric factor that is derived by a local L2-fitting criterion for the regression adjustment estimation. Then the Monte Carlo numerical simulation studies are carried out to show the efficiency and accuracy of the present work. The mean square error (MSE) is a measure of the average squared deviation of the estimated function values from the actual ones. The values of MSE indicate that the innovation in noise structure is performed well in comparison with the existing noise in the nonlinear autoregressive models.
Stochastic Processes and their Applications, 1991
... Sciences, 1053 Budapest, Hungary M. Csorgo** Department of Mathematics and Statistics,Carleton University, Ottawa, Ontario, Canada KIS5B6 ZY Lin*** Department of Mathematics, Hangzhou University, Hangzhou, Zhejiang, People's Republic of China P. R6Vesz**** Institut ...
The statistical analysis for equations driven by fractional Gaussian process (fGp) is obviously 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 $dX_t=\theta X_tdt+dG_t,\ t\geq0$ with an unknown parameter $\theta>0$, where $G$ is a Gaussian process. We provide sufficient conditions, based on the properties of $G$, ensuring the strong consistency and the asymptotic distributions of our estimator $\widetilde{\theta}_t$ of $\theta$ based on the observation $\{X_s,\ s\in[0,t]\}$ as $t\rightarrow\infty$. Our approach offers an elementary, unifying proof of \cite{BEO}, and it allows to extend the result of \cite{BEO} to the case when $G$ is a fractional Brownian motion with Hurst parameter $H\in(0,1)$. We also discuss the cases of subfractional Browian motion and bifractional Brownian motion.
Journal of Statistical Planning and Inference, 2011
In this paper, we investigate the maximum likelihood estimation for the reflected Ornstein-Uhlenbeck (ROU) processes based on continuous observations. Both the cases with one-sided barrier and two-sided barriers are considered. We derive the explicit formulas for the estimators, and then prove their strong consistency and asymptotic normality. Moreover, the bias and mean square errors are represented in terms of the solutions to some PDEs with homogeneous Neumann boundary conditions. We also illustrate the asymptotic behavior of the estimators through a simulation study.
Communications on Stochastic Analysis, 2010
Let X = {X t } be an infinitely divisible stationary process. A good measure of the asymptotic dependence structure of X is provided by the limit of ρ X (t) as t → ∞, where ρ X (t) is equal to the joint characteristic function of (X t , X 0) minus the product of the characteristic functions of X t and X 0. An interesting case is when ρ X (t) → 0; which roughly says that, as time becomes large, the future of the random phenomenon (represented by X) is becoming independent of its past. In this paper, we study the rate of decay of ρ X (t) (as t → ∞) when X is an Ornstein-Uhlenbeck (r, α)-semi-stable process. The results obtained here generalize and complement the corresponding results for Ornstein-Uhlenbeck α-stable and Ornstein-Uhlenbeck (Gaussian) processes.
2016
It is considered a transmittion process of a useful signal in Ornstein-Uhlenbeck model in C[-l,l[ defined by the stochastic differential equation dΨ(t,x,ω)=∑_n=0^2m A_n∂^n/∂ x^nΨ(t,x,ω)dt +σ d W(t,ω) with initial condition Ψ(0,x,ω)=Ψ_0(x) ∈ FD^(0)[-l,l[, where m > 1, (A_n)_0 < n < 2m∈R^+×R^2m-1, ((t,x,ω) ∈ [0,+∞[× [-l,l[ ×Ω), σ∈R^+, C[-l,l[ is Banach space of all real-valued bounded continuous functions on [-l,l[, FD^(0)[-l,l[ ⊂C[-l,l[ is class of all real-valued bounded continuous functions on [-l,l[ whose Fourier series converges to himself everywhere on [-l,l[, (W(t,ω))_t > 0 is a Wiener process and Ψ_0(x) is a useful signal. By use a sequence of transformed signals (Z_k)_k ∈ N=(Ψ(t_0,x,ω_k))_k ∈ N at moment t_0>0, consistent and infinite-sample consistent estimations of the useful signal Ψ_0 is constructed under assumption that parameters (A_n)_0 < n < 2m and σ are known. Animation and simulation of the Ornstein-Uhlenbeck process in C[-l,l[ and an estimation...
An Ornstein-Uhlenbeck (OU) process can be considered as a continuous time interpolation of the discrete time AR$(1)$ process. Departing from this fact, we analyse in this work the effect of iterating OU treated as a linear operator that maps a Wiener process onto Ornstein-Uhlenbeck process, so as to build a family of higher order Ornstein-Uhlenbeck processes, OU$(p)$, in a similar spirit as the higher order autoregressive processes AR$(p)$. We show that for $p \ge 2$ we obtain in general a process with covariances different than those of an AR$(p)$, and that for various continuous time processes, sampled from real data at equally spaced time instants, the OU$(p)$ model outperforms the appropriate AR$(p)$ model. Technically our composition of the OU operator is easy to manipulate and its parameters can be computed efficiently because, as we show, the iteration of OU operators leads to a process that can be expressed as a linear combination of basic OU processes. Using this expression...
Annals of the Institute of Statistical Mathematics, 2012
We study the problem of parameter estimation for Ornstein-Uhlenbeck processes driven by symmetric α-stable motions, based on discrete observations. A least squares estimator is obtained by minimizing a contrast function based on the integral form of the process. Let h be the length of time interval between two consecutive observations. For both the case of fixed h and that of h → 0, consistencies and asymptotic distributions of the estimator are derived. Moreover, for both of the cases of h, the estimator has a higher order of convergence for the Ornstein-Uhlenbeck process driven by non-Gaussian α-stable motions (0 < α < 2) than for the process driven by the classical Gaussian case (α = 2).
Statistical Inference for Stochastic Processes, 2014
In this paper we investigate the large-sample behaviour of the maximum likelihood estimate (MLE) of the unknown parameter θ for processes following the model dξ t = θ f (t)ξ t dt + dB t , where f : R → R is a continuous function with period, say P > 0. Here the periodic function f (•) is assumed known. We establish the consistency of the MLE and we point out its minimax optimality. These results comply with the well-established case of an Ornstein Uhlenbek process when the function f (•) is constant. However the case when P 0 f (t)dt = 0 and f (•) is not identically null presents some special features. For instance in this case whatever is the value of θ , the rate of convergence of the MLE is T as in the case when θ = 0 and P 0 f (t)dt = 0.
Journal of Multivariate Analysis, 1996
It is shown that the suitably normalized maximum likelihood estimators of some parameters of multidimensional Ornstein–Uhlenbeck processes with coefficient matrix of a special structure have exactly a normal distribution. This result provides a generalization to an arbitrary dimension of the well-known behavior of the estimator of the period of a complex AR(1) process.
Statistics & Probability Letters, 2007
Consider the first time an Ornstein-Uhlenbeck process starting from zero crosses a constant positive threshold. Assuming that the asymptotic mean is above the threshold, conditions on the asymptotic variance relative to the distance between the threshold and the asymptotic mean are given that ensures the finiteness of the positive Laplace transforms. r
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