Study of Dependence for Some Stochastic Processes

Abstract

This article is concerned with studying the following problem: Consider a multivariate stochastic process whose law is characterized in terms of some infinitesimal characteristics, such as the infinitesimal generator in case of finite Markov chains. Under what conditions imposed on these infinitesimal characteristics of this multivariate process, the univariate components of the process agree in law with given univariate stochastic processes. Thus, in a sense, we study a stochastic processe' counterpart of the stochastic dependence problem, which in case of real valued random variables is solved in terms of Sklar's theorem.     

The Fractional Calculus for Some Stochastic Processes

Abstract

The integration and differentiation of fractional orders are well known concepts for deterministic functions (see Miller, K.S.; Ross, B. An Introduction to Fractional Calculus and Fractional Differential Equations; John Wiley: New York, 1993; I. Podlubny and Ahmed M.A. El-Sayed, On two definitions of fractional calculus Slovak Academy of Sciences Institute of experimental Phys. UEF-03-96 ISBN 80-7099-252-2, 1996; Podlubny, I. Fractional Differential Equations; Acad. Press: San Diego – New York, London etc. 1999; Samko, S.G.; Kilbas, A.A.; Marichev, O. Integral and derivatives of the fractional orders and some of their applications. Nauka i Teknika Minisk 1983). In earlier work, we have studied the fractional calculus for mean square continuous stochastic processes. In this work, we shall study the mean square (m.s.) fractional calculus for stochastic processes which are m.s. Riemann-integrable and prove some its properties.     

Lp空间中随机过程泛函的某些收敛性

随机变量之和的收敛性问题已有许多人在研究,并取得很多很好的结果.本文则进一步讨论了随机过程之和在Lp空间中依联合测度收敛的情况.     

Statistical Inference for Stochastic Processes

Table of Contents

Statistical Inference for Stochastic Processes     

Approximating Rough Stochastic PDEs

We study approximations to a class of vector‐valued equations of Burgers type driven by a multiplicative space‐time white noise. A solution theory for this class of equations has been developed recently in Probability Theory Related Fields by Hairer and Weber. The key idea was to use the theory of controlled rough paths to give definitions of weak/mild solutions and to set up a Picard iteration argument. In this article the limiting behavior of a rather large class of (spatial) approximations to these equations is studied. These approximations are shown to converge and convergence rates are given, but the limit may depend on the particular choice of approximation. This effect is a spatial analogue to the Itô‐Stratonovich correction in the theory of stochastic ordinary differential equations, where it is well known that different approximation schemes may converge to different solutions.© 2014 Wiley Periodicals, Inc.     

Max-Plus Stochastic Processes

This paper is concerned with processes which are max-plus counterparts of Markov diffusion processes governed by Ito sense stochastic differential equations. Concepts of max-plus martingale and max-plus stochastic differential equation are introduced. The max-plus counterparts of backward and forward PDEs for Markov diffusions turn out to be first-order PDEs of Hamilton–Jacobi–Bellman type. Max-plus additive integrals and a max-plus additive dynamic programming principle are considered. This leads to variational inequalities of Hamilton–Jacobi–Bellman type.     

Disorder Indicator for Nonstationary Stochastic Processes

Doklady Mathematics - The properties of a statistic called a self-consistent stationary level of nonstationary time series are examined. It is shown that a change in this statistic can be treated...     

Maximal Probability Inequalities for Stochastic Processes

Let X_a,b be nonnegative random variables with the property that X_a,b ≦ X_a,c + X_c.b for all 0≦ a < c < b ≦ T, where T > 0 is fixed. We define M_a,b = sup {X_a,c: a < c ≦ h} and establish bounds for P[M_a,b ≧ λ] in terms of given bounds for P[X_a,b ≧ λ], where λ runs through some interval (0, λ_o), 0 < λ_o ≦ ∞ fixed. These bounds explicitly involve a nonnegative function g(a, b) assumed to be quasi-superadditive with an index Q, i.e., g(a, c) + g(c, b) ≦ Qg(a, b) for all 0≦ a < c < b < T, where 1 ≦ Q < 2 is fixed. Maximal inequalities obtained in this way can be applied to stochastic processes exhibiting long-range dependence. Among others, these applications may include certain self-similar processes such as fractional Brownian motion, stochastic processes occurring in linear time series models, etc.     

Statistical Korovkin Theory for Multivariate Stochastic Processes

In this article, we study Korovkin-type approximation theorems for multivariate stochastic processes via the concept of A-statistical convergence. A non-trivial example expressing the importance of our results is also presented.     

Haar-Based Multiresolution Stochastic Processes

Modifying a Haar wavelet representation of Brownian motion yields a class of Haar-based multiresolution stochastic processes in the form of an infinite series $$X_t = \sum_{n=0}^\infty\lambda_n\varDelta _n(t)\epsilon_n,$$ where ?? _n ?? _n (t) is the integral of the nth Haar wavelet from 0 to t, and ?? _n are i.i.d. random variables with mean 0 and variance 1. Two sufficient conditions are provided for X _t to converge uniformly with probability one. Each stochastic process , the collection of all almost sure uniform limits, retains the second-moment properties and the same roughness of sample paths as Brownian motion, yet lacks some of the features of Brownian motion, e.g., does not have independent and/or stationary increments, is not Gaussian, is not self-similar, or is not a martingale. Two important tools are developed to analyze elements of , the nth-level self-similarity of the associated bridges and the tree structure of dyadic increments. These tools are essential in establishing sample path results such as H?lder continuity and fractional dimensions of graphs of the processes.     

Large Deviation Probabilities for Square-Gaussian Stochastic Processes

We investigate properties of square-Gaussian stochastic processes. These processes are formed by quadratic forms of Gaussian processes or by limits in the mean square of quadratic forms of Gaussian processes. Special classes of these processes are determined and investigated. For processes from these classes estimates of large deviation probability are obtained. These estimates we use to estimate the probability that Gaussian vector-valued process leave some region on some interval of time. We construct asymptotic confidence regions for estimates of covariance functions of vector-valued Gaussian processes. Criterion of hypothesis testing on covariance functions of these processes is constructed.     

Integration by Parts for Poisson Processes

Using a perturbation of the rate of a Poisson process and an inverse time change, an integration by parts formula is obtained. This enables a new form of the integrand in a martingale representation result to be obtained.     

关于随机阶的几个结果

在较弱条件下,对独立随机变量序列的极大值序列,给出了几个关于随机阶的结果.     

Call for Papers: Bayesian Inference for Stochastic Processes

Call for Papers

Call for Papers: Bayesian Inference for Stochastic Processes     

Infinite Divisibility for Stochastic Processes and Time Change

General results concerning infinite divisibility, selfdecomposability, and the class L _m property as properties of stochastic processes are presented. A new concept called temporal selfdecomposability of stochastic processes is introduced. Lévy processes, additive processes, selfsimilar processes, and stationary processes of Ornstein–Uhlenbeck type are studied in relation to these concepts. Further, time change of stochastic processes is studied, where chronometers (stochastic processes that serve to change time) and base processes (processes to be time-changed) are independent but do not, in general, have independent increments. Conditions for inheritance of infinite divisibility and selfdecomposability under time change are given.