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We perform a numerical analysis of a class of randomly perturbed Hamiltonian systems and Poisson systems. For the considered additive noise perturbation of such systems, we show the long-time behaviour of the energy and quadratic Casimirs for the exact solution. We then propose and analyse a drift-preserving splitting scheme for such problems with the following properties: exact drift preservation of energy and quadratic Casimirs, mean-square order of convergence 1, weak order of convergence 2. These properties are illustrated with numerical experiments. [ABSTRACT FROM AUTHOR]

In the paper, an universality theorem of discrete type on the approximation of analytic functions by shifts of a special absolutely convergent Dirichlet series is obtained. These series is close in a certain sense to the periodic zeta-function and depends on a parameter. [ABSTRACT FROM AUTHOR]

Let H(D) be the space of analytic functions on the strip D = {σ + it ∈ C : 1/2 < σ < 1}. In this paper, it is proved that there exists a closed non-empty set Fα1,...,αr ⊂ H(D) such that every collection of the functions (f1, . . ., fr) ∈ Fα1,...,αr is approximated by discrete shifts (ζ(s + ikh1, α1), . . ., ζ(s + ikhr, αr)), hj > 0, j = 1, . . ., r, k ∈ N ∪ {0}, of Hurwitz zeta-functions with arbitrary parameters α1, . . ., αr. [ABSTRACT FROM AUTHOR]

The class of balanced implicit methods (BIMs) has been introduced by Milstein, Platen and Schurz (1998) for numerical integration of d -dimensional systems of ordinary Itô-type stochastic differential equations (SDEs) driven by independent Wiener processes. They also established the rate r = 0.5 of mean square convergence towards exact solution along equidistant partitions under appropriate assumptions. The other major convergence criterion is that of weak convergence (i.e. the convergence of moments E [ f (X) ]) of them with respect to certain test classes of functions f and remains to be an open question. We fill that gap by proving weak convergence of a subclass of BIMs with rate r = 1.0 with respect to the test class of C 2 -functions and with variable step sizes. This class of stochastic-numerical approximations represents a linear-implicit family of schemes stabilizing p -th moments by a computationally efficient implementation. The novel proof relies on a random version of Clarkson-Beckner inequality to estimate its p -th moments. Moreover, we prove stability and boundedness of a more general class of these methods in any dimension d. [ABSTRACT FROM AUTHOR]

We provide scaling limits for the block counting process and the fixation line of Λ-coalescents as the initial state n tends to infinity under the assumption that the measure Λ on [0, 1] satisfies ∫[0,1] u-1|Λ - bλ|(du) < ∞ for some b ≥ 0. Here λ denotes the Lebesgue measure on [0, 1]. The main result states that the block counting process, properly transformed, converges in the Skorohod space to a generalized Ornstein-Uhlenbeck process as n tends to infinity. The result is applied to beta coalescents with parameters 1 and b > 0. We split the generators into two parts by additively decomposing Λ into a 'Bolthausen-Sznitman part' bλ and a 'dust part' Λ - bλ and then prove the uniform convergence of both parts separately. [ABSTRACT FROM AUTHOR]

We study Markov chains on ℤm, m ≥ 2, that behave like a standard symmetric random walk outside of the hyperplane (membrane) H = {0} × ℤm−1. The exit probabilities from the membrane (penetration probabilities) H are periodic and also depend on the incoming direction to H, what makes the membrane H two-sided. Moreover, sliding along the membrane is allowed. We show that the natural scaling limit of such Markov chains is a m-dimensional diffusion whose first coordinate is a skew Brownian motion and the other m − 1 coordinates is a Brownian motion with a singular drift controlled by the local time of the first coordinate at 0. In the proof we utilize a martingale characterization of the Walsh Brownian motion and determine the effective permeability and slide direction. Eventually, a similar convergence theorem is established for the one-sided membrane without slides and random iid penetration probabilities. [ABSTRACT FROM AUTHOR]

This paper deals with a split equality hierarchical fixed point problem in real Hilbert spaces which is an important and natural extension of hierarchical fixed point problem and split equality fixed point problem. An iterative algorithm where the stepsizes do not depend on the operator norms, so called simultaneous Krasnoselski-Mann algorithm is suggested for solving the split equality hierarchical fixed point problem. Further we prove a weak convergence theorem for the sequence generated by this algorithm. This special aspect of the algorithm together with the convergence result makes it an interesting scheme. Furthermore, we give some examples to justify the main result. Finally, we show that our purposed iterative algorithm is more efficient than some other known iterative algorithms. On the other hand, the framework is general and allows us to treat in a unified way several iterative algorithms, recovering, developing and improving some recently known related convergence results in the literature. [ABSTRACT FROM AUTHOR]

In the paper, we consider the multidimensional renewal equation in matrix form. The renewal equation for the process with independent increments and states of the Markov process is found. The renewal function is investigated. The limit theorem for the renewal equation is proved. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Zhejiang University (Science Edition) is the property of Journal of Zhejiang University (Science Edition) Editorial Office and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The problem of optimal control at finite time interval for a system of differential equations with impulse action at fixed moments of time as well as the corresponding averaged system of ordinary differential equations are considered. It is proved the existence of optimal control of exact and averaged problems. Also, it is established that optimal control of averaged problem realize the approximate optimal synthesis of exact problem. The main result of the article is a theorem, where it is proved that optimal contol of an averaged problem is almost optimal for exact problem. Substantiation of proximity of solutions of exact and averaged problems is obtained.

Conditional Value-at-Risk (CoVaR) is defined as the Value-at-Risk of a certain risk given that the related risk equals a given threshold (CoVaR=) or is smaller/larger than a given threshold (CoVaR

Let H(D) be the space of analytic functions on the strip . In this paper, it is proved that there exists a closed non-empty set such that every collection of the functions is approximated by discrete shifts , of Hurwitz zeta-functions with arbitrary parameters

In the paper, an universality theorem of discrete type on the approximation of analytic functions by shifts of a special absolutely convergent Dirichlet series is obtained. These series is close in a certain sense to the periodic zeta-function and depends on a parameter.

We show that geodesic random walks on a complete Finsler manifold of bounded geometry converge to a diffusion process which is, up to a drift, the Brownian motion corresponding to a Riemannian metric. [ABSTRACT FROM AUTHOR]

M-eigenvalues of fourth-order partially symmetric tensors play an important role in nonlinear elasticity and materials. In this paper, we present some M-eigenvalue intervals to locate all M-eigenvalues of fourth-order partially symmetric tensors. It is proved that the new interval is tighter than the one proposed by He, Li and Wei [ 16 ]. Furthermore, we obtain some new checkable sufficient conditions for the strong ellipticity of fourth-order partially symmetric tensors. Three numerical examples arising from anisotropic materials are presented to verify the efficiency of the proposed results. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce a new inertial iterative method for solving split variational inclusion problems in real Hilbert spaces. We prove that the generated sequence converges weakly to the solution of the considered problem under some mild conditions. The major contributions of our results are: (ⅰ) to increase the rate of convergence of the method for solving split variational inclusion problem through the inertial extrapolation step, (ⅱ) to relax the choice of the inertial factor and show the inertial factor can be chosen greater than 1/3 unlike what is previously known before for inertial proximal point method in the literature (ⅲ) to show the numerical efficiency and superiority of our proposed method through some test example. [ABSTRACT FROM AUTHOR]

In this paper, an iterative method with alternated inertial extrapolation step is proposed to solve the split feasibility problem. The stepsize in the proposed algorithm uses the self adaptive technique, which does not depend on the prior information of the operator norm. The weak convergence is proved under suitable conditions. Finally, a numerical example is given to illustrate the effectiveness of our algorithm. [ABSTRACT FROM AUTHOR]

In this work, we investigate pseudomonotone variational inequality problems in a real Hilbert space and propose two projection-type methods with inertial terms for solving them. The first method does not require prior knowledge of the Lipschitz constant and the second one does not require the Lipschitz continuity of the mapping which governs the variational inequality. A weak convergence theorem for our first algorithm is established under pseudomonotonicity and Lipschitz continuity assumptions, and a weak convergence theorem for our second algorithm is proved under pseudomonotonicity and uniform continuity assumptions. We also establish a nonasymptotic O(1/n) convergence rate for our proposed methods. In order to illustrate the computational effectiveness of our algorithms, some numerical examples are also provided. [ABSTRACT FROM AUTHOR]

We consider a multidimensional normally or obliquely reflected diffusion in a smooth domain. We approximate it by solutions of stochastic differential equations without reflection using the penalty method, that is, we approximate the reflection term with an additional drift term. In the existing literature, usually a specific approximating sequence is provided to prove the existence of a reflected diffusion. In this paper, we provide general sufficient conditions on the approximating coefficients. [ABSTRACT FROM AUTHOR]

In this work, we study the partial sums of independent and identically distributed random variables with the number of terms following a fractional Poisson (FP) distribution. The FP sum contains the Poisson and geometric summations as particular cases. We show that the weak limit of the FP summation, when properly normalized, is a mixture between the normal and Mittag-Leffler distributions, which we call by Normal-Mittag-Leffler (NML) law. A parameter estimation procedure for the NML distribution is developed and the associated asymptotic distribution is derived. Simulations are run to check the performance of the proposed estimators under finite samples. An empirical illustration on the daily log-returns of the Brazilian stock exchange index (IBOVESPA) shows that the NML distribution captures better the tails than some of its competitors. Related problems such as a mixed Poisson representation for the FP law and the weak convergence for the Conway-Maxwell-Poisson random sum are also addressed. [ABSTRACT FROM AUTHOR]

We consider the modified Surface Quasi-Geostrophic (mSQG) equation on the 2D torus T 2 , perturbed by multiplicative transport noise. The equation admits the white noise measure on T 2 as the invariant measure. We first prove the existence of white noise solutions to the stochastic equation via the method of point vortex approximation, then, under a suitable scaling limit of the noise, we show that the solutions converge weakly to the unique stationary solution of the dissipative mSQG equation driven by space–time white noise. The weak uniqueness of the latter equation is also proved by following Gubinelli and Perkowski's approach in Gubinelli and Perkowski (2020). [ABSTRACT FROM AUTHOR]

Stochastic models of chemical reaction networks are an important tool to describe and analyze noise effects in cell biology. When chemical species and reaction rates in a reaction system have different orders of magnitude, the associated stochastic system is often modeled in a multiscale regime. It is known that multiscale models can be approximated with a reduced system such as mean field dynamics or hybrid systems, but the accuracy of the approximation remains unknown. In this paper, we estimate the probability distribution of low copy species in multiscale stochastic reaction systems under a short timescale. We also establish an error bound for this approximation. Throughout the paper, typical mass-action systems are mainly handled, but we also show that the main theorem can be extended to general kinetics, which generalizes existing results in the literature. Our approach is based on a direct analysis of the Kolmogorov equation, in contrast to classical approaches in the existing literature. [ABSTRACT FROM AUTHOR]

Copyright of Afrika Statistika is the property of Statistics & Probability African Society and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

A theorem on the convergence of a random process generated by a nearly critical branching process with immigration to a deterministic process is proved in this article. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the numerical solutions to index 1 stochastic differential algebraic equations. We introduce a new class of weak second-order stochastic Runge–Kutta methods for finding the numerical approximate solutions to multi-dimensional stochastic differential algebraic equations. A four-stage stiffly accurate stochastic Runge–Kutta methods for approximating analytical solutions to index 1 stochastic differential algebraic equations are derived. By colored rooted tree analysis, the order conditions for the stochastic Runge–Kutta methods of order two satisfying the weak convergence is obtained. The scalar test equations are considered to obtain the mean-square stability and the T-stability of weak second-order stochastic Runge–Kutta methods. Finally, some numerical illustrations are provided to prove the theoretical findings.

The Lerch zeta function is defined by a Dirichlet series depending on two fixed parameters. In the paper, we consider the approximation of analytic functions by discrete shifts of the Lerch zeta function, and we prove that, for arbitrary parameters and a step of arithmetic progression, there is a closed non-empty subset of the space of analytic functions defined in the critical strip such that its functions can be approximated by discrete shifts of the Lerch zeta function. The set of those shifts is infinite, and it has a positive density. For the proof, the weak convergence of probability measures in the space of analytic functions is applied.

Let tτ be a solution to the equation θ(t)=(τ−1)π, τ>0, where θ(t) is the increment of the argument of the function π−s/2Γ(s/2) along the segment connecting points s=1/2 and s=1/2+it. tτ is called the Gram function. In the paper, we consider the approximation of collections of analytic functions by shifts of the Riemann zeta-function (ζ(s+itτα1),…,ζ(s+itταr)), where α1,…,αr are different positive numbers, in the interval [T,T+H] with H=o(T), T→∞, and obtain the positivity of the density of the set of such shifts. Moreover, a similar result is obtained for shifts of a certain absolutely convergent Dirichlet series connected to ζ(s). Finally, an example of the approximation of analytic functions by a composition of the above shifts is given.

Let u = {u(t,x),t ∈ [0,T],x ∈ ℝ} be a solution to a stochastic heat equation driven by a space-time white noise. We study that the realized power variation of the process u with respect to the time, properly normalized, has Gaussian asymptotic distributions. In particular, we study the realized power variation of the process u with respect to the time converges weakly to Brownian motion. [ABSTRACT FROM AUTHOR]

In many operations management settings, the arrival process to the system exhibits clear nonstationarities. These nonstationarities may arise as a consequence of time-of-day effects, day-of-week effects, seasonalities, or stochastic fluctuations in the arrival rate. Tools that are developed for performance prediction and decision making when systems are stationary may not be valid in the nonstationary environment. This paper provides a generic, closed-form, and simple approximation tool for when the nonstationarities are caused by rapid fluctuations in the arrival processes. This paper introduces a new asymptotic regime for simplifying stochastic models having nonstationary effects, such as those that arise in the presence of time-of-day effects. This regime describes an operating environment within which the arrival process to a service system has an arrival intensity that is fluctuating rapidly. We show that such a service system is well approximated by the corresponding model in which the arrival process is Poisson with a constant arrival rate. In addition to the basic weak convergence theorem, we also establish a first order correction for the distribution of the cumulative number of arrivals over [ 0 , t ] , as well as the number-in-system process for an infinite-server queue fed by an arrival process having a rapidly changing arrival rate. This new asymptotic regime provides a second regime within which nonstationary stochastic models can be reasonably approximated by a process with stationary dynamics, thereby complementing the previously studied setting within which rates vary slowly in time. [ABSTRACT FROM AUTHOR]

A Freidlin–Wentzell type large deviation principle is established for stochastic partial differential equations with slow and fast time-scales, where the slow component is a one-dimensional stochastic Burgers equation with small noise and the fast component is a stochastic reaction-diffusion equation. Our approach is via the weak convergence criterion developed in [A. Budhiraja and P. Dupuis, A variational representation for positive functionals of infinite dimensional Brownian motion, Probab. Math. Statist. 20 (2000) 39–61]. [ABSTRACT FROM AUTHOR]

This work addresses some asymptotic behavior of solutions to stochastic convective Brinkman–Forchheimer (SCBF) equations perturbed by multiplicative Gaussian noise in two and three dimensional bounded domains. Using a weak convergence approach of Budhiraja and Dupuis, we establish the Laplace principle for the strong solution to SCBF equations in a suitable Polish space. Then, the Wentzell–Freidlin type large deviation principle is derived using the well known results of Varadhan and Bryc. The large deviations for short time are also considered in this work. Furthermore, we study the exponential estimates on certain exit times associated with the solution trajectory of SCBF equations. Using contraction principle, we study these exponential estimates of exit times from the frame of reference of Wentzell–Freidlin type large deviations principle (LDP). This work also improves several LDP results available in the literature for tamed Navier–Stokes equations as well as Navier–Stokes equations with damping in bounded domains. [ABSTRACT FROM AUTHOR]

The split feasibility problem is to find a point x∗ with the property that x∗∈ C and Ax∗∈ Q, where C and Q are nonempty closed convex subsets of real Hilbert spaces X and Y, respectively, and A is a bounded linear operator from X to Y. The split feasibility problem models inverse problems arising from phase retrieval problems and the intensity-modulated radiation therapy. In this paper, we introduce a new inertial relaxed CQ algorithm for solving the split feasibility problem in real Hilbert spaces and establish weak convergence of the proposed CQ algorithm under certain mild conditions. Our result is a significant improvement of the recent results related to the split feasibility problem. [ABSTRACT FROM AUTHOR]

We study the limit distribution of partial sums with a random number of terms following a class of mixed Poisson distributions. The resulting weak limit is a mixture between a normal distribution and an exponential family, which we call by normal exponential family (NEF) laws. A new stability concept is introduced and a relationship between α -stable distributions and NEF laws is established. We propose the estimation of the NEF model parameters through the method of moments and also by the maximum likelihood method via an Expectation–Maximization algorithm. Monte Carlo simulation studies are addressed to check the performance of the proposed estimators, and an empirical illustration of the financial market is presented. [ABSTRACT FROM AUTHOR]

A theorem is obtained on the approximation of a collection of analytic functions in short intervals by a collection of shifts of the Riemann zeta-function , where are algebraic numbers linearly independent over the field of rational numbers. [ABSTRACT FROM AUTHOR]

We consider Borel measures on separable Banach spaces that are limits of their finite-dimensional images in the weak topology. The class of Banach spaces on which all measures have this property is introduced. The specified property is proved for all measures from the closure in variation of the linear span of the set of measures absolutely continuous with respect to Gaussian measures. Connections with the approximation property and the stochastic approximation property are considered. [ABSTRACT FROM AUTHOR]

In this paper, we first study a relaxed inertial projection and contraction method for variational inequalities to obtain weak convergence results under standard assumptions in Hilbert spaces. Next, we propose another inertial projection and contraction method for which the inertial factor θ is chosen in [ 0 , 1 ] with θ = 1 possible. Weak and linear convergence are obtained for this second proposed method. As far as we know, the choice θ = 1 has not been considered before in the literature for inertial projection and contraction method. Finally, we give some numerical illustrations of our methods. [ABSTRACT FROM AUTHOR]

We give a particle picture interpretation of two recently discovered classes of self-similar stable processes with stationary increments studied by Samorodnitsky et al. and Dombry and Guillotin-Plantard. We study the occupation times of certain Poissonian systems of particles with ±1 charges and i.i.d. heavy-tailed weights, moving independently according to Lévy processes. We also obtain a new class of self-similar stable processes. [ABSTRACT FROM AUTHOR]

Let C be a nonempty subset of a Banach space X. A mapping T: C → C is said to satisfy(Bγ,μ) condition if there exists γ ∈ [0,1] and μ ∈ [0, ½] satisfying 2μ ≤ γ such that for each x,y ∈ C, γ||x-Tx||≤||x-y||+μ||y-Ty|| implies ||Tx-Ty||≤(1-γ)||x-y||+μ(||x-Ty||+||y-Tx||). In this paper, we obtain some convergence theorems for such mappings using M iterative process in uniformly convex Banach space setting. Our results extend and improve many results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we show an approximation in law, in the space of continuous functions on [0, I]2, of two-parameter Gaussian processes that can be represented as a Wiener type integral by processes constructed from processes that converge to the Brownian sheet. As an application, we obtain a sequence of processes constructed from a Lévy sheet that converges in law towards the fractional Brownian sheet. [ABSTRACT FROM AUTHOR]

The Kuramoto--Sivashinsky equation is a nonlinear parabolic partial differential equation, which describes the instability and turbulence of waves in chemical reactions and laminar flames. The aim of this work is to prove the large deviation principle for the stochastic Kuramoto--Sivashinsky equation driven by multiplicative noise. To establish the large deviation principle, the weak convergence approach is used, which relies on proving basic qualitative properties of controlled versions of the original stochastic partial differential equation. [ABSTRACT FROM AUTHOR]

In the paper, an absolutely convergent Dirichlet series whose shifts approximate a wide class of analytic functions is constructed. This series is close in the mean to the Riemann zeta-function. [ABSTRACT FROM AUTHOR]

The Kuramoto–Sivashinsky equation is a nonlinear parabolic partial differential equation, which describes the instability and turbulence of waves in chemical reactions and laminar flames. The aim of this work is to prove the large deviation principle for the stochastic Kuramoto–Sivashinsky equation driven by multiplicative noise. To establish the large deviation principle, the weak convergence approach is used, which relies on proving basic qualitative properties of controlled versions of the original stochastic partial differential equation.