<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.     

股价波动的指数O-U过程模型

本文针对股价波动的几何布朗运动模型对收益率假设的缺陷,对该模型进行了改进,并建立了股价波动的指数OkU过程模型,得到了比传统模型更好的结果.     

The properties of connected graphs and some corrections to the list of Uhlenbeck and Ford

Some graph-theoretical results for connected cluster integrals are used to correct six entries in the table of Uhlenbeck and Ford.(1)     

Parameter Estimation with Exact Distribution for Multidimensional Ornstein–Uhlenbeck Processes

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.     

Infinite dimensional Ornstein–Uhlenbeck processes with unbounded diffusion – Approximation,quadratic variation,and Itô formula

The paper studies a class of Ornstein–Uhlenbeck processes on the classical Wiener space. These processes are associated with a diffusion type Dirichlet form whose corresponding diffusion operator is unbounded in the Cameron–Martin space. It is shown that the distributions of certain finite dimensional Ornstein–Uhlenbeck processes converge weakly to the distribution of such an infinite dimensional Ornstein–Uhlenbeck process. For the infinite dimensional processes, the ordinary scalar quadratic variation is calculated. Moreover, relative to the stochastic calculus via regularization, the scalar as well as the tensor quadratic variation are derived. A related Itô formula is presented.     

On Harnack inequality for <Emphasis Type="Italic">α</Emphasis>-stable Ornstein–Uhlenbeck processes

We consider the α-stable Ornstein–Uhlenbeck process in with the generator . We show that if 2 > α ≥ 1 or α < 1 = d the Harnack inequality holds. For α < 1 < d we construct a counterexample that shows that the Harnack inequality does not hold. Partially supported by KBN and MEN.     

Small ball probabilities for Gaussian random fields and tensor products of compact operators

We find the logarithmic -small ball asymptotics for a large class of zero mean Gaussian fields with covariances having the structure of ``tensor product'. The main condition imposed on marginal covariances is the regular behavior of their eigenvalues at infinity that is valid for a multitude of Gaussian random functions including the fractional Brownian sheet, Ornstein - Uhlenbeck sheet, etc. So we get the far-reaching generalizations of well-known results by Csáki (1982) and by Li (1992). Another class of Gaussian fields considered is the class of additive fields studied under the supremum-norm by Chen and Li (2003). Our theorems are based on new results on spectral asymptotics for the tensor products of compact self-adjoint operators in Hilbert space which are of independent interest.

     

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.