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In this present work, we study new algorithms to solve the split equality fixed-point problems for total quasi-asymptotically nonexpansive mappings in Hilbert spaces. In addition, we have established the convergence criteria for the proposed algorithms, at the end, we provided numerical results that justified our theoretical results. The results presented in this paper, provided a unified framework for studying this kind of problem involving different classes of mappings. [ABSTRACT FROM AUTHOR]

We consider stochastic differential equations with bounded α -Hölder continuous drift, with α ∈ (0 , 1) , driven by multiplicative noise. We moreover assume that the diffusion coefficient is uniformly elliptic and bounded, with bounded first and second order partial derivatives. We show that the weak rate of convergence for the Euler–Maruyama scheme is arbitrarily close to (1 + α) / 2. [ABSTRACT FROM AUTHOR]

We analyse convergence of a micro–macro acceleration method for the simulation of stochastic differential equations with time-scale separation. The method alternates short bursts of path simulations with the extrapolation of macroscopic state variables forward in time. After extrapolation, a new microscopic state is constructed, consistent with the extrapolated macroscopic state, that minimises the perturbation caused by the extrapolation in a relative entropy sense. We study local errors and numerical stability of the method to prove its convergence to the full microscopic dynamics when the extrapolation time step tends to zero and the number of macroscopic state variables tends to infinity. [ABSTRACT FROM AUTHOR]

In this paper we show that solutions of two-dimensional stochastic Navier–Stokes equations driven by Brownian motion can be approximated by stochastic Navier–Stokes equations forced by pure jump noise/random kicks. [ABSTRACT FROM AUTHOR]

In traditional works on numerical schemes for solving stochastic differential equations (SDEs), the globally Lipschitz assumption is often assumed to ensure different types of convergence. In practice, this is often too strong a condition. Brownian motion driven SDEs used in applications sometimes have coefficients which are only Lipschitz on compact sets, but the paths of the SDE solutions can be arbitrarily large. In this paper, we prove convergence in probability and a weak convergence result under a less restrictive assumption, that is, locally Lipschitz and with no finite time explosion. We prove if a numerical scheme converges in probability uniformly on any compact time set (UCP) with a certain rate under a global Lipschitz condition, then the UCP with the same rate holds when a globally Lipschitz condition is replaced with a locally Lipschitz plus no finite explosion condition. For the Euler scheme, weak convergence of the error process is also established. The main contribution of this paper is the proof of n weak convergence of the normalized error process and the limit process is also provided. We further study the boundedness of the second moments of the weak limit process and its running supremum under both global Lipschitz and locally Lipschitz conditions. [ABSTRACT FROM AUTHOR]

Abstract A fundamental algorithm for selecting ranks from a finite subset of an ordered set is Radix Selection. This algorithm requires the data to be given as strings of symbols over an ordered alphabet, e.g., binary expansions of real numbers. Its complexity is measured by the number of symbols that have to be read. In this paper the model of independent data identically generated from a Markov chain is considered. The complexity is studied as a stochastic process indexed by the set of infinite strings over the given alphabet. The orders of mean and variance of the complexity and, after normalization, a limit theorem with a centered Gaussian process as limit are derived. This implies an analysis for two standard models for the ranks: uniformly chosen ranks, also called grand averages, and the worst case rank complexities which are of interest in computer science. For uniform data and the asymmetric Bernoulli model (i.e. memoryless sources), we also find weak convergence for the normalized process of complexities when indexed by the ranks while for more general Markov sources these processes are not tight under the standard normalizations. [ABSTRACT FROM AUTHOR]

Abstract In applications the properties of a stochastic feature often change gradually rather than abruptly, that is: after a constant phase for some time they slowly start to vary. In this paper we discuss statistical inference for the detection and the localization of gradual changes in the jump characteristic of a discretely observed Ito semimartingale. We propose a new measure of time variation for the jump behaviour of the process. The statistical uncertainty of a corresponding estimate is analysed by deriving new results on the weak convergence of a sequential empirical tail integral process and a corresponding multiplier bootstrap procedure. [ABSTRACT FROM AUTHOR]

This article deals with the weak error in averaging principle for a stochastic wave equation on a bounded interval [ 0 , L ] , perturbed by an oscillating term arising as the solution of a stochastic reaction–diffusion equation evolving on the fast time scale. Under suitable conditions, it is proved that the rate of weak convergence of the original solution to the solution of the corresponding averaged equation is of order 1 via an asymptotic expansion approach. [ABSTRACT FROM AUTHOR]

We provide conditions on dependent and on non-stationary random variables X n ensuring that the mantissa of the sequence of products ∏ 1 n X k is almost surely distributed following Benford’s law or converges in distribution to Benford’s law. This is achieved through proving new generalizations of Lévy’s and Robbins’s results on distribution modulo 1 of sums of independent random variables. [ABSTRACT FROM AUTHOR]

We study a class of non-stationary shot noise processes which have a general arrival process of noises with non-stationary arrival rate and a general shot shape function. Given the arrival times, the shot noises are conditionally independent and each shot noise has a general (multivariate) cumulative distribution function (c.d.f.) depending on its arrival time. We prove a functional weak law of large numbers and a functional central limit theorem for this new class of non-stationary shot noise processes in an asymptotic regime with a high intensity of shot noises, under some mild regularity conditions on the shot shape function and the conditional (multivariate) c.d.f. We discuss the applications to a simple multiplicative model (which includes a class of non-stationary compound processes and applies to insurance risk theory and physics) and the queueing and work-input processes in an associated non-stationary infinite-server queueing system. To prove the weak convergence, we show new maximal inequalities and a new criterion of existence of a stochastic process in the space D given its consistent finite dimensional distributions, which involve a finite set function with the superadditive property. [ABSTRACT FROM AUTHOR]

This paper shows how the theory of Dirichlet forms can be used to deliver proofs of optimal scaling results for Markov chain Monte Carlo algorithms (specifically, Metropolis–Hastings random walk samplers) under regularity conditions which are substantially weaker than those required by the original approach (based on the use of infinitesimal generators). The Dirichlet form methods have the added advantage of providing an explicit construction of the underlying infinite-dimensional context. In particular, this enables us directly to establish weak convergence to the relevant infinite-dimensional distributions. [ABSTRACT FROM AUTHOR]

Given an Itō semimartingale with a time-homogeneous jump part observed at high frequency, we prove weak convergence of a normalized truncated empirical distribution function of the Lévy measure to a Gaussian process. In contrast to competing procedures, our estimator works for processes with a non-vanishing diffusion component and under simple assumptions on the jump process. [ABSTRACT FROM AUTHOR]

In this paper, a weak Local Linearization scheme for Stochastic Differential Equations (SDEs) with multiplicative noise is introduced. First, for a time discretization, the solution of the SDE is locally approximated by the solution of the piecewise linear SDE that results from the Local Linearization strategy. The weak numerical scheme is then defined as a sequence of random vectors whose first moments coincide with those of the piecewise linear SDE on the time discretization. The scheme is explicit, preserves the first two moments of the solution of SDEs with linear drift and diffusion coefficients in state and time, and inherits the mean-square stability or instability that such solution may have. The rate of convergence is derived and numerical simulations are presented for illustrating the performance of the scheme. [ABSTRACT FROM AUTHOR]

Consider the n th iterated Brownian motion I ( n ) = B n ∘ ⋯ ∘ B 1 . Curien and Konstantopoulos proved that for any distinct numbers t i ≠ 0 , ( I ( n ) ( t 1 ) , … , I ( n ) ( t k ) ) converges in distribution to a limit I [ k ] independent of the t i ’s, exchangeable, and gave some elements on the limit occupation measure of I ( n ) . Here, we prove under some conditions, finite dimensional distributions of n th iterated two-sided stable processes converge, and the same holds the reflected Brownian motions. We give a description of the law of I [ k ] , of the finite dimensional distributions of I ( n ) , as well as those of the iterated reflected Brownian motion iterated ad libitum. [ABSTRACT FROM AUTHOR]

In this paper we study a conditional version of the Wang transform in the context of discrete GARCH models and their diffusion limits. Our first contribution shows that the conditional Wang transform and Duans generalized local risk-neutral valuation relationship based on equilibrium considerations, lead to the same GARCH option pricing model. We derive the weak limit of an asymmetric GARCH model risk-neutralized via Wang’s transform. The connection with stochastic volatility limits constructed using other standard pricing kernels, such as the conditional Esscher transform or the extended Girsanov principle, is further investigated by comparing the corresponding market prices of variance risk. [ABSTRACT FROM AUTHOR]

Let X be a Markov process taking values in E with continuous paths and transition function ( P s , t ) . Given a measure μ on ( E , E ) , a Markov bridge starting at ( s , ε x ) and ending at ( T ∗ , μ ) for T ∗ < ∞ has the law of the original process starting at x at time s and conditioned to have law μ at time T ∗ . We will consider two types of conditioning: (a) weak conditioning when μ is absolutely continuous with respect to P s , t ( x , ⋅ ) and (b) strong conditioning when μ = ε z for some z ∈ E . The main result of this paper is the representation of a Markov bridge as a solution to a stochastic differential equation (SDE) driven by a Brownian motion in a diffusion setting. Under mild conditions on the transition density of the underlying diffusion process we establish the existence and uniqueness of weak and strong solutions of this SDE. [ABSTRACT FROM AUTHOR]

To avoid finding the stationary distributions of stochastic differential equations by solving the nontrivial Kolmogorov–Fokker–Planck equations, the numerical stationary distributions are used as the approximations instead. This paper is devoted to approximate the stationary distribution of the underlying equation by the Backward Euler–Maruyama method. Currently existing results (Mao et al., 2005; Yuan et al., 2005; Yuan et al., 2004) are extended in this paper to cover larger range of nonlinear SDEs when the linear growth condition on the drift coefficient is violated. [ABSTRACT FROM AUTHOR]

We consider a large declining population of cells under an external selection pressure, modeled as a subcritical branching process. This population has genetic variation introduced at a low rate which leads to the production of exponentially expanding mutant populations, enabling population escape from extinction. Here we consider two possible settings for the effects of the mutation: Case (I) a deterministic mutational fitness advance and Case (II) a random mutational fitness advance. We first establish a functional central limit theorem for the renormalized and sped up version of the mutant cell process. We establish that in Case (I) the limiting process is a trivial constant stochastic process, while in Case (II) the limit process is a continuous Gaussian process for which we identify the covariance kernel. Lastly we apply the functional central limit theorem and some other auxiliary results to establish a central limit theorem (in the large initial population limit) of the first time at which the mutant cell population dominates the population. We find that the limiting distribution is Gaussian in both Cases (I) and (II), but a logarithmic correction is needed in the scaling for Case (II). This problem is motivated by the question of optimal timing for switching therapies to effectively control drug resistance in biomedical applications. [ABSTRACT FROM AUTHOR]

Abstract: This article uses variational method for studying existence and uniqueness of solutions for impulsive evolution equations. The main techniques include Hilbert triple, Sobolev embedding theorem, Galerkin approximation and weak convergence for passing to the limit. [Copyright &y& Elsevier]

We consider a slow–fast stochastic differential system with Lévy noise. We will employ the perturbed test function method to study the normal deviation of the slow–fast system. Our main result states that the deviation can be approximated by a Gaussian process and the central limit theorem is obtained for the system. [ABSTRACT FROM AUTHOR]