Almost sure exponential stability of backward Euler-Maruyama discretizations for hybrid stochastic differential equations

This is a continuation of the first author’s earlier paper [1] jointly with Pang and Deng, in which the authors established some sufficient conditions under which the Euler-Maruyama (EM) method can reproduce the almost sure exponential stability of the test hybrid SDEs. The key condition imposed in [1] is the global Lipschitz condition. However, we will show in this paper that without this global Lipschitz condition the EM method may not preserve the almost sure exponential stability. We will then show that the backward EM method can capture the almost sure exponential stability for a certain class of highly nonlinear hybrid SDEs.     

Central Limit Theorem Started at a Point for?Stationary Processes and Additive Functionals of?Reversible Markov Chains

In this paper we study the almost sure central limit theorem started at a point for additive functionals of a stationary and ergodic Markov chain via a martingale approximation in the almost sure sense. Some of the results provide sufficient conditions for general stationary sequences. We use these results to study the quenched CLT for additive functionals of reversible Markov chains.     

Exponential stability of impulsive stochastic functional differential equations

In this paper, we investigate the pth moment and almost sure exponential stability of impulsive stochastic functional differential equations with finite delay by using Lyapunov method. Several stability theorems of impulsive stochastic functional differential equations with finite delay are derived. These new results are employed to impulsive stochastic equations with bounded time-varying delays and stochastically perturbed equations. Meanwhile, an example and simulations are given to show that impulses play an important role in pth moment and almost sure exponential stability of stochastic functional differential equations with finite delay.     

Self-normalized limit theorems for linear processes generated by <Emphasis Type="Italic">ρ</Emphasis>-mixing innovations<Superscript>*</Superscript>

In this paper, we study the asymptotic behavior of the self-normalizer V _n ² for partial sums of linear processes generated by strictly stationary ρ-mixing innovations with infinite variance. Further, by using this we derive self-normalized versions of the CLT, the functional CLT, and the almost sure CLT for partial sums of the processes.     

Large and Moderate Deviations Principles and Central Limit Theorem for the Stochastic 3D Primitive Equations with Gradient-Dependent Noise

Journal of Theoretical Probability - We establish the large deviations principle (LDP), the moderate deviations principle (MDP), and an almost sure version of the central limit theorem (CLT) for...     

An elementary approach to Brownian local time based on simple,symmetric random walks

Summary In this paper we define Brownian local time as the almost sure limit of the local times of a nested sequence of simple, symmetric random walks. The limit is jointly continuous in <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>(t,x)$. The rate of convergence is $n^{\frac14} (\log n)^{\frac34}$ that is close to the best possible. The tools we apply are almost exclusively from elementary probability theory.     

When is a linear combination of independent fBm’s equivalent to a single fBm?

We study and answer the question posed in the title. The answer is derived from some new necessary and sufficient conditions for equivalence of Gaussian processes with stationary increments and recent frequency domain results for the fBm. The result shows in particular precisely in which cases the local almost sure behaviour of a linear combination of independent fBm’s is the same as that of a multiple of a single fBm.     

A universal result in almost sure central limit theory

The discovery of the almost sure central limit theorem (Brosamler, Math. Proc. Cambridge Philos. Soc. 104 (1988) 561–574; Schatte, Math. Nachr. 137 (1988) 249–256) revealed a new phenomenon in classical central limit theory and has led to an extensive literature in the past decade. In particular, a.s. central limit theorems and various related ‘logarithmic’ limit theorems have been obtained for several classes of independent and dependent random variables. In this paper we extend this theory and show that not only the central limit theorem, but every weak limit theorem for independent random variables, subject to minor technical conditions, has an analogous almost sure version. For many classical limit theorems this involves logarithmic averaging, as in the case of the CLT, but we need radically different averaging processes for ‘more sensitive’ limit theorems. Several examples of such a.s. limit theorems are discussed.     

一类连续Gauss过程的拟必然q变差

以双分数次Brown运动为例,本文对一类具有较弱性质的连续Gauss过程x证明其q变差2n-1∑i=0 |x((i+1)2-n)∧t-X(i2-n)∧t|q拟必然收敛到0.对双参数情形我们也给出相应的结果.     

Brown运动增量拟必然局部Strassen重对数律

该文建立了Brown运动增量的拟必然局部Strassen重对数律.利用这一结果,得到了Brown运动拟必然泛函连续模.     

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We establish a quenched central limit theorem (CLT) for the branching Brownian motion with random immigration in dimension $d\geq4$. The limit is a Gaussian random measure, which is the same as the annealed central limit theorem, but the covariance kernel of the limit is different from that in the annealed sense when d=4.     

带Markov调制的随机微分延迟方程的稳定性

本文采用了一例特定的Lyapunov函数,来研究带Markov调制的随机微分延迟方程的p阶指数稳定性,并对其几乎必然指数稳定性也进行了探讨.     

Razumikhin-type exponential stability criteria of neutral stochastic functional differential equations

The paper discusses both pth moment and almost sure exponential stability of solutions to neutral stochastic functional differential equations and neutral stochastic differential delay equations, by using the Razumikhin-type technique. The main goal is to find sufficient stability conditions that could be verified more easily then by using the usual method with Lyapunov functionals. The analysis is based on paper [X. Mao, Razumikhin-type theorems on exponential stability of neutral stochastic functional differential equations, SIAM J. Math. Anal. 28 (2) (1997) 389-401], referring to mean square and almost sure exponential stability.     

Limit Theorems for Logarithmic Averages of Fractional Brownian Motions

We prove almost sure invariance principles for logarithmic averages of fractional Brownian motions.Research supported byResearch supported by     

Path Collapse for an Inhomogeneous Random Walk

On an open interval we follow the paths of a Brownian motion which returns to a fixed point as soon as it reaches the boundary and restarts afresh indefinitely. We determine that two paths starting at different points either cannot collapse or they do so almost surely. The problem can be modelled as a spatially inhomogeneous random walk on a group and contrasts sharply with the higher dimensional case in that if two paths may collapse they do so almost surely.     

A problem of Földes and Puri on the Wiener process

Let be a real-valued Wiener process starting from 0, and be the right-continuous inverse process of its local time at 0. Földes and Puri [3] raise the problem of studying the almost sure asymptotic behavior of as tends to infinity, i.e. they ask: how long does stay in a tube before ``crossing very much" a given level? In this note, both limsup and liminf laws of the iterated logarithm are provided for .

     

Intersections of Brownian motions

This article presents a survey of the theory of the intersections of Brownian motion paths. Among other things, we present a truly elementary proof of a classical theorem of A. Dvoretzky, P. Erdős and S. Kakutani. This proof is motivated by old ideas of P. Lévy that were originally used to investigate the curve of planar Brownian motion.     

Almost sure central limit theorems on the Wiener space

In this paper, we study almost sure central limit theorems for sequences of functionals of general Gaussian fields. We apply our result to non-linear functions of stationary Gaussian sequences. We obtain almost sure central limit theorems for these non-linear functions when they converge in law to a normal distribution.     

Almost-sure path properties of fractional Brownian sheet

The almost sure sample function behavior of the vector-valued fractional Brownian sheet is investigated. In particular, the global and the local moduli of continuity of the sample functions are studied. These results give precise information about the continuity and the oscillation behavior of the sample functions.