Irregular Sampling of Generalized Harmonizable Processes

Abstract

In this note an irregular sampling expansion for bandlimited harmonizable processes is obtained by employing contour integral techniques in the vector-valued analytic functions setting. In so doing, we use the integral representation of a harmonizable process with respect to a vector measure.     

Absolutely Summing Processes

Abstract

Absolutely summing processes are defined, which form a subclass of weakly operator harmonizable processes. When the parameter space is the set of real numbers, it is proved that an absolutely summing process is represented as an integral of operator stationary processes with respect to an appropriate probability measure. To do this, weak convergence of scalar and vector measures is considered. Then we prove compactness of the unit ball of vector measures under certain topologies, and we apply the Choquet theorem to derive an integral representation.     

A classification of vector harmonizable processes

Hilbert space valued second order stochastic processes over the real line are considered. Various hmonizabilities and V-boundednesses are introduced and their interrelations are obtained as well as heir integral representations. Examples are given to distinguish most of the harmonizabilities. Stationary dilations of harmonizable processes are also discussed. Finally, for some harmonizabilities, some convergence theorems for sequences of processes are obtained     

Stochastic integration for operator valued processes on hilbert spaces and on nuclear spaces

The representation of a nuclear space valued square integrable martingale by means of another nuclear space valued square integrable martingale is given in terms of stochastic inegrals of operator valued processes. The construction of the stochastic integral goes through that of operator valued processes on Hilbert spaces. A new approach is given for the Hilbertian case, so that only the integration of Hilbert-Schmidt operator valued processes is needed to represent square integrable martingales     

Simulation of stochastic integrals with respect to Lévy processes of type G

We study the simulation of stochastic processes defined as stochastic integrals with respect to type G Lévy processes for the case where it is not possible to simulate the type G process exactly. The type G Lévy process as well as the stochastic integral can on compact intervals be represented as an infinite series. In a practical simulation we must truncate this representation. We examine the approximation of the remaining terms with a simpler process to get an approximation of the stochastic integral. We also show that a stochastic time change representation can be used to obtain an approximation of stochastic integrals with respect to type G Lévy processes provided that the integrator and the integrand are independent.     

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.      

Stochastic integral representation theorem for quantum semimartingales

The quantum stochastic integral of Itô type formulated by Hudson and Parthasarathy is extended to a wider class of adapted quantum stochastic processes on Boson Fock space. An Itô formula is established and a quantum stochastic integral representation theorem is proved for a class of unbounded semimartingales which includes polynomials and (Wick) exponentials of the basic martingales in quantum stochastic calculus.     

Stochastic processes indexed by hypergroups. I

For many applications it is desirable to have extensions of the classical theory of weakly stationary processes to certain classes of nonstationary ones. A large family for which Fourier methods still play a major role is the harmonizable class and some related processes. The purpose of this paper is to initiate the study of a class of stochastic processes with a stationarity condition based on the notion of hypergroups.     

线性可移算子与可调和的随机过程

<正> §1.引言T.Onoyama 利用了 Weyl-Stone-Titchmarsh 的特征函数(Eigenfunction)展开公式,对具有二阶矩的实的连续机过程(本文中所用的极限,系指在均方意义下的极限)求得了下列隨机函数方程     

A Series Approach to Stochastic Differential Equations with Infinite Dimensional Noise

This paper considers semilinear stochastic differential equations in Hilbert spaces with Lipschitz nonlinearities and with the noise terms driven by sequences of independent scalar Wiener processes (Brownian motions). The interpretation of such equations requires a stochastic integral. By means of a series of Itô integrals, an elementary and direct construction of a Hilbert space valued stochastic integral with respect to a sequence of independent scalar Wiener processes is given. As an application, existence and strong and weak uniqueness for the stochastic differential equation are shown by exploiting the series construction of the integral.     

A Poisson bridge between fractional Brownian motion and stable Lévy motion

We study a non-Gaussian and non-stable process arising as the limit of sums of rescaled renewal processes under the condition of intermediate growth. The process has been characterized earlier by the cumulant generating function of its finite-dimensional distributions. Here, we derive a more tractable representation for it as a stochastic integral of a deterministic function with respect to a compensated Poisson random measure. Employing the representation we show that the process is locally and globally asymptotically self-similar with fractional Brownian motion and stable Lévy motion as its tangent limits.     

Stochastic integration for Lévy processes with values in Banach spaces

A stochastic integral of Banach space valued deterministic functions with respect to Banach space valued Lévy processes is defined. There are no conditions on the Banach spaces or on the Lévy processes. The integral is defined analogously to the Pettis integral. The integrability of a function is characterized by means of a radonifying property of an integral operator associated with the integrand. The integral is used to prove a Lévy–Itô decomposition for Banach space valued Lévy processes and to study existence and uniqueness of solutions of stochastic Cauchy problems driven by Lévy processes.     

Backward stochastic Volterra integral equations—Representation of adapted solutions

For backward stochastic Volterra integral equations (BSVIEs, for short), under some mild conditions, the so-called adapted solutions or adapted M-solutions uniquely exist. However, satisfactory regularity of the solutions is difficult to obtain in general. Inspired by the decoupling idea of forward–backward stochastic differential equations, in this paper, for a class of BSVIEs, a representation of adapted M-solutions is established by means of the so-called representation partial differential equations and (forward) stochastic differential equations. Well-posedness of the representation partial differential equations are also proved in certain sense.     

Multiple stochastic integrals constructed by special expansions of products of the integrating stochastic processes

The paper deals with problems of constructing multiple stochastic integrals in the case when the product of increments of the integrating stochastic process admits an expansion as a finite sum of series with random coefficients. This expansion was obtained for a sufficiently wide class including centered Gaussian processes. In the paper, some necessary and sufficient conditions are obtained for the existence of multiple stochastic integrals defined by an expansion of the product of Wiener processes. It was obtained a recurrent representation for the Wiener stochastic integral as an analog of the Hu–Meyer formula.     

On set-valued stochastic integrals in an M-type 2 Banach space

In a separable Banach space, for set-valued martingale, several equivalent conditions based on the measurable selections are discussed, and then, in an M-type 2 Banach space, at first we define single valued stochastic integral by the differential of a real valued Brownian motion, after that extend it to set-valued case. We prove that the set-valued stochastic integral becomes a set-valued submartingale, which is different from single valued case, and obtain the Castaing representation theorem for the set-valued stochastic integral, which is applicable for set-valued stochastic differential equations.     

Cylindrical fractional Brownian motion in Banach spaces

In this article we introduce cylindrical fractional Brownian motions in Banach spaces and develop the related stochastic integration theory. Here a cylindrical fractional Brownian motion is understood in the classical framework of cylindrical random variables and cylindrical measures. The developed stochastic integral for deterministic operator valued integrands is based on a series representation of the cylindrical fractional Brownian motion, which is analogous to the Karhunen–Loève expansion for genuine stochastic processes. In the last part we apply our results to study the abstract stochastic Cauchy problem in a Banach space driven by cylindrical fractional Brownian motion.     

New classes of self-similar symmetric stable random fields

We construct two new classes of symmetric stable self-similar random fields with stationary increments, one of the moving average type, the other of the harmonizable type. The fields are defined through an integral representation whose kernel involves a norm on ⁿ . We examine how the choice of the norm affects the finite-dimensional distributions. We also study the processes which are obtained by projecting the random fields on a one-dimensional subspace. We compare these projection processes with each other and with other well-known self-similar processes and we characterize their asymptotic dependence structure.The research was done at Boston University while the first author was on leave from the Hugo Steinhaus Center, Poland. The second author was partially supported by the ONR Grant N00014-90-J-1287 at Boston University and by a grant of the United States-Israel Binational Science Foundation.     

关于白噪声的L_2－泛函在Malliavin算子作用下的积分表示

本文基于非线性空间的张量积结构，建立了抽象可测空间（ＴＳＦＨＢ，）上关于白噪声测度Ｘ的（非适应）随机积分．应用Ｃｈａｏｓ分解，得出了关于白噪声的一般Ｌ２－泛函ξ的如下积分表示式　进一步，我们讨论了所建立的随机积分在Ｍａｌｌｉａｎ算子Ｌ作用下的特点，从而获得ξ在Ｌ作用下的如下随机积分表示     

Tempered Fractional Stable Motion

Tempered fractional stable motion adds an exponential tempering to the power-law kernel in a linear fractional stable motion, or a shift to the power-law filter in a harmonizable fractional stable motion. Increments from a stationary time series that can exhibit semi-long-range dependence. This paper develops the basic theory of tempered fractional stable processes, including dependence structure, sample path behavior, local times, and local nondeterminism.