Non-Differentiable Skew Convolution Semigroups and Related Ornstein–Uhlenbeck Processes

It is proved that a general non-differentiable skew convolution semigroup associated with a strongly continuous semigroup of linear operators on a real separable Hilbert space can be extended to a differentiable one on the entrance space of the linear semigroup. A càdlàg strong Markov process on an enlargement of the entrance space is constructed from which we obtain a realization of the corresponding Ornstein–Uhlenbeck process. Some explicit characterizations of the entrance spaces for special linear semigroups are given.     

On a Dirichlet Problem Involving an Ornstein–Uhlenbeck Operator

We consider an elliptic Dirichlet problem which involves Ornstein–Uhlenbeck operators of special form in a half space of R ⁿ . We obtain necessary and sufficient conditions under which global Schauder estimates in spaces of Hölder continuous and bounded functions hold. For this purpose we use analytical tools, in particular semigroups and interpolation theory. Moreover we extend a theorem on the analiticity of subordinated semigroups (see Carasso and Kato; Trans. Amer. Math. Soc. 327 (1990, 867–877)) to a class of Markov type semigroups. We also provide explicit formulas for the Poisson kernels.     

Branching Processes with Immigration and Related Topics

This is a survey on the recent progresses in the study of branching processes with immigration, generalized Ornstein-Uhlenbeck processes, and affine Markov processes. We mainly focus on the applications of skew convolution semigroups and the connections in those processes.     

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.     

Optimal bond portfolios with fixed time to maturity

We study interest rate models where the term structure is given by an affine relation and in particular where the driving stochastic processes are so-called generalized Ornstein–Uhlenbeck processes.     

测度值分枝过程与移民过程

本文介绍了测度值分枝过程和由斜卷积半群定义的伴随移民过程的基本理论和研究现状,主要内容包括：分枝粒子系统的收敛;超过程的基本正则性和极限定理;非线性微分方程;广义分枝模型;斜卷积半群和进入律;用Ｋｕｚｎｅｔｓｏｖ过程构造移民过程等。     

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.     

Stochastic Perturbations of Line Solitons of KP

We consider the properties of localized solutions of the KP equation coupled to a stochastic noise. Corresponding to white noise, we find that the traveling waves are destroyed asymptotically, and we determine the distribution of the wave position and the arrival time. For generalized Ornstein–Uhlenbeck processes, we show that the only effect of noise is to render the asymptotic position random; in particular, when the noise has a sufficiently strong attenuation mechanism, the random wave coincides asymptotically with the unperturbed one. We also consider linearization of the corresponding Cauchy problem in the plane corresponding to this kind of initial data.     

Higher-order Riesz operators for the Ornstein–Uhlenbeck Semigroup

We prove that the second-order Riesz transforms associated to the Ornstein–Uhlenbeck semigroup are weak type (1,1) with respect to the Gaussian measure in finite dimension. We also show that they are given by a principal value integral plus a constant multiple of the identity. For the Riesz transforms of order three or higher, we present a counterexample showing that the weak type (1,1) estimate fails.     

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.     

Least squares estimator for discretely observed Ornstein–Uhlenbeck processes with small Lévy noises

We study the problem of parameter estimation for generalized Ornstein–Uhlenbeck processes with small Lévy noises, observed at n regularly spaced time points on [0, 1]. Least squares method is used to obtain an estimator of the drift parameter. The consistency and the rate of convergence of the least squares estimator (LSE) are established when a small dispersion parameter ε→0 and n→∞ simultaneously. The asymptotic distribution of the LSE in our general setting is shown to be the convolution of a normal distribution and a stable distribution. The obtained results are different from the classical cases where asymptotic distributions are normal.     

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 Sharp Constant in the Small Ball Asymptotics of Some Gaussian Processes under L 2-Norm

We find the sharp constant in the small L ₂-deviation asymptotics for a wide class of Gaussian processes including the m-times integrated Wiener process and the m-times integrated Ornstein–Uhlenbeck process. Extremal properties of usual and Euler integration are proved. Bibliography: 19 titles.     

Generalized z-Distributions and Related Stochastic Processes

The class of generalized z–distributions is defined and their properties are investigated. Ornstein–Uhlenbeck–type and self–similar generalized z–processes are constructed and described. Esscher transforms of the generalized z–processes and the mixed generalized z–processes are characterized. Finally, construction and some properties of generalized z–diffusions are also discussed.     

Mehler semigroups,Ornstein-Uhlenbeck processes and background driving Lévy processes on locally compact groups and on hypergroups

For finite dimensional vector spaces it is well-known that there exists a 1-1-correspondence between distributions of Ornstein-Uhlenbeck type processes (w.r.t. a fixed group of automorphisms) and (background driving) Lévy processes, hence between M- or skew convolution semigroups on the one hand and continuous convolution semigroups on the other. An analogous result could be proved for simply connected nilpotent Lie groups. Here we extend this correspondence to a class of commutative hypergroups.     

Retarded Stationary Ornstein–Uhlenbeck Processes Driven by Lévy Noise and Operator Self-Decomposability

A class of Langevin equations driven by Lévy processes with time delays are considered. Sufficient conditions are established to find a unique stationary solution of functional stochastic systems studied. The concept of operator self-decomposability, closely related to the stationary solutions, is generalized to retarded Ornstein–Uhlenbeck processes so as that useful conditions under which random variables with self-decomposability are embedded into a stationary retarded Langevin equations are found.     

The Class of Distributions of Periodic Ornstein–Uhlenbeck Processes Driven by Lévy Processes

The class I(c) of stationary distributions of periodic Ornstein–Uhlenbeck processes with parameter c driven by Lévy processes is analyzed. A characterization of I(c) analogous to a well-known characterization of the selfdecomposable distributions is given. The relations between I(c) for varying values of c and the relations with the class of selfdecomposable distributions and with the nested classes L_m are discussed.     

Markov Processes Related with Dunkl Operators

Dunkl operators are parameterized differential-difference operators on ^Nthat are related to finite reflection groups. They can be regarded as a generalization of partial derivatives and play a major role in the study of Calogero–Moser–Sutherland-type quantum many-body systems. Dunkl operators lead to generalizations of various analytic structures, like the Laplace operator, the Fourier transform, Hermite polynomials, and the heat semigroup. In this paper we investigate some probabilistic aspects of this theory in a systematic way. For this, we introduce a concept of homogeneity of Markov processes on ^Nthat generalizes the classical notion of processes with independent, stationary increments to the Dunkl setting. This includes analogues of Brownian motion and Cauchy processes. The generalizations of Brownian motion have the càdlàg property and form, after symmetrization with respect to the underlying reflection groups, diffusions on the Weyl chambers. A major part of the paper is devoted to the concept of modified moments of probability measures on ^Nin the Dunkl setting. This leads to several results for homogeneous Markov processes (in our extended setting), including martingale characterizations and limit theorems. Furthermore, relations to generalized Hermite polynomials, Appell systems, and Ornstein–Uhlenbeck processes are discussed.     

Banach Space-Valued Ornstein–Uhlenbeck Processes with General Drift Coefficients

We construct Ornstein–Uhlenbeck processes with values in Banach space and with continuous paths. The drift coefficient must only generate a strongly continuous semigroup on the Hilbert space which determines the Brownian motion. We admit arbitrary starting points and consider also invariant measures for the process, generalizing earlier work in many directions. A price for the generality is that sometimes one has to enlarge the phase space but most previously known results are covered.The constructions are based on abstract Wiener space methods, more precisely on images of abstract Wiener spaces under suitable linear transformations of the Cameron–Martin space. The image abstract Wiener measures are then given by stochastic extensions. We present the basic spaces and operators and the most important results on image spaces and stochastic extensions in some detail.     

A Diffusion Approximation for a Markovian Queue with Reneging

Consider a single-server queue with a Poisson arrival process and exponential processing times in which each customer independently reneges after an exponentially distributed amount of time. We establish that this system can be approximated by either a reflected Ornstein–Uhlenbeck process or a reflected affine diffusion when the arrival rate exceeds or is close to the processing rate and the reneging rate is close to 0. We further compare the quality of the steady-state distribution approximations suggested by each diffusion.