Transition Law-based Simulation of Generalized Inverse Gaussian Ornstein–Uhlenbeck Processes

In this paper, a stochastic integral of Ornstein–Uhlenbeck type is represented to be the sum of three independent random variables—one follows a distribution whose density is a deconvolution of the densities of two generalized inverse Gaussian distributions, and the two others all have compound Poisson distributions. Based on the representation of the stochastic integral, a simulation procedure for obtaining discretely observed values of Ornstein–Uhlenbeck processes with given generalized inverse Gaussian distribution is provided. For some subclasses of the generalized inverse Gaussian Ornstein–Uhlenbeck process, the innovations can be sampled exactly. The performance of the simulation method is evidenced by some empirical results.     

On large deviations in testing Ornstein–Uhlenbeck-type models

We obtain exact large deviation rates for the log-likelihood ratio in testing models with observed Ornstein–Uhlenbeck processes and get explicit rates of decrease for the error probabilities of Neyman–Pearson, Bayes, and minimax tests. Moreover, we give expressions for the rates of decrease for the error probabilities of Neyman–Pearson tests in models with observed processes solving affine stochastic delay differential equations.      

Poissonian subordinators,the Wiener–Ornstein–Uhlenbeck field,and a relation between the Ornstein–Uhlenbeck processes and Brownian bridges

We apply to a sequence of i.i.d. random variables a time change operator via a Poisson process that is independent of this sequence. We consider sums of independent copies of processes constructed in this way and having continuous time. Finite limit distributions of these sums coincide with the finite limit distributions of the Wiener–Ornstein–Uhlenbeck field that is the tensor product of a Brownian motion and the Ornstein–Uhlenbeck process. The transition characteristics of the limit Ornstein–Uhlenbeck process are described by Brownian bridges that are builded into the Wiener–Ornstein–Uhlenbeck field. Bibliography: 4 titles.     

On the coupling property and the Liouville theorem for Ornstein–Uhlenbeck processes

Using a coupling for the weighted sum of independent random variables and the explicit expression of the transition semigroup of Ornstein–Uhlenbeck processes driven by compound Poisson processes, we establish the existence of a successful coupling and the Liouville theorem for general Ornstein–Uhlenbeck processes. Then we present the explicit coupling property of Ornstein–Uhlenbeck processes directly from the behaviour of the corresponding symbol or characteristic exponent. This approach allows us to derive gradient estimates for Ornstein–Uhlenbeck processes via the symbol.     

Exact Simulation of IG-OU Processes

IG-OU processes are a subclass of the non-Gaussian processes of Ornstein–Uhlenbeck type, which are important models appearing in financial mathematics and elsewhere. The simulation of these processes is of interest for its applications in statistical inference. In this paper, a stochastic integral of Ornstein–Uhlenbeck type is represented to be the sum of two independent random variables—one has an inverse Gaussian distribution and the other has a compound Poisson distribution. And in distribution, the compound Poisson random variable is equal to a sum of Poisson-distributed number positive random variables, which are independent identically distributed and have a common specified density function. The exact simulation of the IG-OU processes, proceeding from time 0 and going in steps of time interval Δ, is achieved via the representation of the stochastic integral. Comparing to the approximate method, which is based on Rosinski’s infinite series representation of the same stochastic integral, by the quantile–quantile plots, the advantage of the exact simulation method is obvious. In addition, as an application, we provide an estimator of the intensity parameter of the IG-OU processes and validate its superiority to another estimator by our exact simulation method.      

Sequential Estimation of the Parameters in a Trigonometric Regression Model with the Gaussian Coloured Noise

This paper considers the estimation problem for a trigonometric regression model with the noise specified by the Ornstein–Uhlenbeck process with unknown parameter. We propose a sequential procedure which ensures a prescribed mean square precision uniformly in the nuisance parameter. The asymptotic behaviour of the procedure duration mean has been studied. This revised version was published online in August 2006 with corrections to the Cover Date.     

Brownian Motion and Ornstein–Uhlenbeck Processes in Planar Shape Space

We discuss Brownian motion and Ornstein–Uhlenbeck processes specified directly in planar shape space. In particular, we obtain the drift and diffusion coefficients of Brownian motion in terms of Kendall shape variables and Goodall–Mardia polar shape variables. Stochastic differential equations are given and the stationary distributions are obtained. By adding in extra drift to a reference figure, Ornstein–Uhlenbeck processes can be studied, for example with stationary distribution given by the complex Watson distribution. The triangle case is studied in particular detail, and some simulations given. Connections with existing work are made, in particular with the diffusion of Euclidean shape. We explore statistical inference for the parameters in the model with an application to cell shape modelling.      

Central and Functional Central Limit Theorems for a Class of Urn Models

We consider an approach based on tails to certain central limit and functional central limit theorems for a class of two color urn models. In particular, some of the results are derived from an associated Ornstein–Uhlenbeck process, and for another result we give an alternative proof based on martingale tails.      

Conditions of smoothness for the distribution density of a solution of a multidimensional linear stochastic differential equation with lévy noise

We obtain a sufficient condition of smoothness for the distribution density of a multidimensional Ornstein–Uhlenbeck process with Lévy noise, i.e., for the solution of a linear stochastic differential equation with Lévy noise.     

Sample path large deviations for a family of long-range dependent traffic and associated queue length processes

We consider the long-range dependent cumulative traffic generated by the superposition of constant rate fluid sources having exponentially distributed inter start times and Pareto distributed durations with finite mean and infinite variance. We prove a sample path large deviation principle when the session start time intensity is increased and the processes are centered and scaled appropriately. Properties of the rate function are investigated. We derive a sample path large deviation principle for a related family of stationary queue length processes. The large deviation approximation of the steady-state queue length distribution is compared with the corresponding empirical distribution obtained by a computer simulation. MSC 2000 Classifications: Primary 60F10; Secondary 60K25, 68M20, 90B22     

Diffusions on path and loop spaces: existence, finite dimensional approximation and Hölder continuity

Summary. We construct Ornstein–Uhlenbeck processes and more general diffusion processes on path and loop spaces of Riemannian manifolds by finite dimensional approximation. We also show H?lder continuity of the sample paths w.r.t. the supremum norm. The proofs are based on the Lyons–Zheng decomposition. Received: 6 September 1996 / In revised form: 1 April 1997     

Properties of the first passage times of the reflected O-U process with a two-sided barrier

In this paper, we give the Laplace transform of the first passage times and obtain the analytic expression of its mean for the reflected Ornstein–Uhlenbeck process with a two-sided barrier for general coefficients.     

On the excursion theory for linear diffusions

We present a number of important identities related to the excursion theory of linear diffusions. In particular, excursions straddling an independent exponential time are studied in detail. Letting the parameter of the exponential time tend to zero it is seen that these results connect to the corresponding results for excursions of stationary diffusions (in stationary state). We characterize also the laws of the diffusion prior and posterior to the last zero before the exponential time. It is proved using Krein’s representations that, e.g. the law of the length of the excursion straddling an exponential time is infinitely divisible. As an illustration of the results we discuss the Ornstein–Uhlenbeck processes.     

Matrix-exponential groups and Kolmogorov–Fokker–Planck equations

Aim of this paper is to provide new examples of H?rmander operators L{\mathcal{L}} to which a Lie group structure can be attached making L{\mathcal{L}} left invariant. Our class of examples contains several subclasses of operators appearing in literature and arising both in theoretical and in applied fields: evolution Kolmogorov operators, degenerate Ornstein–Uhlenbeck operators, Mumford and Fokker–Planck operators, Ornstein–Uhlenbeck operators with time-dependent periodic coefficients. Our examples basically come from exponential of matrices, as well as from linear constant-coefficient ODE’s, in \mathbbR{\mathbb{R}} or in \mathbbC{\mathbb{C}} . Furthermore, we describe how these groups can be combined together to obtain new structures and new operators, also having an interest in the applied field.     

Cox Point Processes Driven by Ornstein–Uhlenbeck Type Processes

The paper is devoted to the development of Cox point processes driven by nonnegative processes of Ornstein–Uhlenbeck (OU) type. Starting with multivariate temporal processes we develop formula for the cross pair correlation function. Further filtering problem is studied by means of two different approaches, either with discretization in time or through the point process densities with respect to the Poisson process. The first approach is described mainly analytically while in the second case we obtain numerical solution by means of MCMC. The Metropolis–Hastings birth–death chain for filtering can be also used when estimating the parameters of the model. In the second part we try to develop spatial and spatio-temporal Cox point processes driven by a stationary OU process. The generating functional of the point process is derived which enables evaluation of basic characteristics. Finally a simulation algorithm is given and applied.      

Local asymptotic mixed normality for discretely observed non-recurrent Ornstein–Uhlenbeck processes

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.     

<Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>-spectrum of Banach space valued Ornstein–Uhlenbeck operators

We characterize the L ¹(E,μ _∞)-spectrum of the Ornstein–Uhlenbeck operator , where μ _∞ is the invariant measure for the Ornstein–Uhlenbeck semigroup generated by L. The main result covers the general case of an infinite-dimensional Banach space E under the assumption that the point spectrum of A ^* is nonempty and extends several recent related results.     

Functional calculus for some perturbations of the Ornstein–Uhlenbeck operator

In a recent paper García-Cuerva et al. have shown that for every p in (1,∞) the symmetric finite-dimensional Ornstein–Uhlenbeck operator has a bounded holomorphic functional calculus on L ^p in the sector of angle . We prove a similar result for some perturbations of the Ornstein–Uhlenbeck operator. Work partially supported by the Progetto Cofinanziato MIUR “Analisi Armonica” and the Gruppo Nazionale INdAM per l’Analisi Matematica, la Probabilitàe le loro Applicazioni.     

Crossing Probabilities for Diffusion Processes with Piecewise Continuous Boundaries

We propose an approach to compute the boundary crossing probabilities for a class of diffusion processes which can be expressed as piecewise monotone (not necessarily one-to-one) functionals of a standard Brownian motion. This class includes many interesting processes in real applications, e.g., Ornstein–Uhlenbeck, growth processes and geometric Brownian motion with time dependent drift. This method applies to both one-sided and two-sided general nonlinear boundaries, which may be discontinuous. Using this approach explicit formulas for boundary crossing probabilities for certain nonlinear boundaries are obtained, which are useful in evaluation and comparison of various computational algorithms. Moreover, numerical computation can be easily done by Monte Carlo integration and the approximation errors for general boundaries are automatically calculated. Some numerical examples are presented.      

Statistical Analysis of the Fractional Ornstein–Uhlenbeck Type Process

We consider the fractional analogue of the Ornstein–Uhlenbeck process, that is, the solution of a one-dimensional homogeneous linear stochastic differential equation driven by a fractional Brownian motion in place of the usual Brownian motion. The statistical problem of estimation of the drift and variance parameters is investigated on the basis of a semimartingale which generates the same filtration as the observed process. The asymptotic behaviour of the maximum likelihood estimator of the drift parameter is analyzed. Strong consistency is proved and explicit formulas for the asymptotic bias and mean square error are derived. Preparing for the analysis, a change of probability method is developed to compute the Laplace transform of a quadratic functional of some auxiliary process. This revised version was published online in August 2006 with corrections to the Cover Date.