On the Infinitesimal Generators of Ornstein–Uhlenbeck Processes with Jumps in Hilbert Space

We study Hilbert space valued Ornstein–Uhlenbeck processes (Y(t), t ≥ 0) which arise as weak solutions of stochastic differential equations of the type dY = JY + CdX(t) where J generates a C ₀ semigroup in the Hilbert space H, C is a bounded operator and (X(t), t ≥ 0) is an H-valued Lévy process. The associated Markov semigroup is of generalised Mehler type. We discuss an analogue of the Feller property for this semigroup and explicitly compute the action of its generator on a suitable space of twice-differentiable functions. We also compare the properties of the semigroup and its generator with respect to the mixed topology and the topology of uniform convergence on compacta.      

An asymptotic test for quantitative gene detection

The problem of detecting the presence of a quantitative gene using a great number of markers in a backcross genetic scheme is addressed.An asymptotic test based on the maximum of a differentiable stochastic process is constructed. Bounds for threshold and power calculation are presented. Simulations and numerical experiments illustrate the convergence towards the asymptotic distribution and the sharpness of the bounds.     

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.      

Properties of the Reflected Ornstein–Uhlenbeck Process

Consider an Ornstein–Uhlenbeck process with reflection at the origin. Such a process arises as an approximating process both for queueing systems with reneging or state-dependent balking and for multi-server loss models. Consequently, it becomes important to understand its basic properties. In this paper, we show that both the steady-state and transient behavior of the reflected Ornstein–Uhlenbeck process is reasonably tractable. Specifically, we (1) provide an approximation for its transient moments, (2) compute a perturbation expansion for its transition density, (3) give an approximation for the distribution of level crossing times, and (4) establish the growth rate of the maximum process.     

Convergence of integrated superpositions of Ornstein-Uhlenbeck processes to fractional Brownian motion

Superpositions of Ornstein-Uhlenbeck processes provide convenient ways to build stationary processes with given marginal distributions and long range dependence. After reviewing some of the basic features, we present several examples of processes with non-Gaussian marginal distributions. Our main results concern asymptotic properties of sums and partial sums of these processes and their polynomial functions. Further, we discuss some applications to estimation.     

Almost Sure Comparisons for First Passage Times of Diffusion Processes through Boundaries

Conditions on the boundary and parameters that produce ordering in the first passage time distributions of two different diffusion processes are proved making use of comparison theorems for stochastic differential equations. Three applications of interest in stochastic modeling are presented: a sensitivity analysis for diffusion models characterized by means of first passage times, the comparison of different diffusion models where first passage times represent an important feature and the determination of upper and lower bounds for first passage time distributions.     

Principles of Large Deviations for the Empirical Processes of the Ornstein–Uhlenbeck Process

The present paper deals with principles of large deviations for the empirical processes of the Ornstein–Uhlenbeck process. One such principle due to Donsker and Varadhan is well known. It uses as underlying space C(, ^d ) endowed with the topology of uniform convergence on compact sets. The principles of large deviations proved in the present paper use as underlying spaces appropriate subspaces of C(, ^d ) endowed with weighted supremum norms. These principles are natural generalizations of the principle of Donsker and Varadhan.     

A Diffusion Equation from the Relativistic Ornstein–Uhlenbeck Process

We derive, in the hydrodynamic limit (large space and time scales), an evolution equation for the particle density in physical space from the (special) relativistic Ornstein–Uhlenbeck process introduced by Debbasch, Mallick, and Rivet. This equation turns out to be identical with the classical diffusion equation, without any relativistic correction. We prove that, in the hydrodynamic limit, this result is indeed compatible with special relativity.     

Local asymptotic mixed normality of transformed Gaussian models for random fields

Local asymptotic mixed normality (LAMN) of a class of transformed Gaussian models for discretely observed random fields is proved. The original Gaussian random field is assumed to be the product of a deterministic process and a process with independent increments. The transformed process is observed only on discrete lattice points in the unit cube and fixed domain asymptotics is investigated. This model is useful for modeling random fields with non-Gaussian marginal distributions.     

Exact <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">2</Emphasis></Subscript>-small ball behavior of integrated Gaussian processes and spectral asymptotics of boundary value problems

We find the exact small deviation asymptotics for the L₂-norm of various m-times integrated Gaussian processes closely connected with the Wiener process and the Ornstein – Uhlenbeck process. Using a general approach from the spectral theory of linear differential operators we obtain the two-term spectral asymptotics of eigenvalues in corresponding boundary value problems. This enables us to improve the recent results from [15] on the small ball asymptotics for a class of m-times integrated Wiener processes. Moreover, the exact small ball asymptotics for the m-times integrated Brownian bridge, the m-times integrated Ornstein – Uhlenbeck process and similar processes appear as relatively simple examples illustrating the developed general theory.Partially supported by grants of RFBR 01-01-00245 and 02-01-01099.