随机微分方程分步单支theta方法的稳定性(英文)

本文研究随机微分方程单支theta方法的均方稳定性.首先,对线性检验方程,当0≤θ<1时,分步单支theta方法在一定的步长限制下能保持原系统的均方稳定性,当θ=1时,方法按任意步长都能保持原系统的稳定性.其次,对满足单边Lipschitz条件的非线性随机微分方程,当1/2<θ0<θ<1时,方法能保持原系统的均方指数稳定性,但对步长有限制,如果θ=1,对步长限制消失.     

带Poisson跳年龄相关随机时滞种群系统的均方稳定性

本文研究了带Poisson跳年龄相关随机时滞种群系统均方稳定性的问题.在一定条件下,给出了数值解均方稳定的定义.利用补偿随机θ法讨论系统数值解的均方稳定性,给出数值解稳定的充分条件.获得了当1/2≤θ≤1时,对于任意的步长?τ/m,数值解是均方稳定的;当0≤θ1,时,如果步长?t∈(0,?t0),数值解是指数均方稳定的的结果.最后通过数值算例推广并验证了结果的有效性和正确性.     

一类随机时滞微分方程随机θ方法的均方收敛率

给出了一类随机时滞微分方程随机θ方法的均方收敛率,这类方程对于时滞项可以不满足Lipschitz条件而仅需要满足一定条件的Hlder连续.     

单调型条件下随机微分方程θ-方法的均方指数稳定性

本文研究高度非线性随机微分方程(SDEs)的数值解稳定性性质.给出θ-方法均方指数稳定性的充分条件.与现有文献不同,本文无需单边线性增长条件和充分小的步长.本文在单调型的条件下,并且至于要步长满足一个很弱的条件即可.因此本文是对现有文献的很大改进.     

非线性随机Pantograph微分方程及其θ-方法的均方渐近稳定性

本文主要研究了非线性随机Pantograph微分方程,讨论了其零解的均方渐近稳定性并给出了零解均方渐近稳定的充分条件.在本文的第三部分,我们将随机θ-方法应用于这类问题,获得了数值解均方渐近稳定条件.     

线性随机比例方程的渐近均方稳定性

本文目的是研究线性随机比例方程解析解和数值方法（连续θ方法）的渐近均方稳定性．给出了解析解和数值方法渐近均方稳定的条件．     

非线性随机延迟微分方程Heun方法的数值稳定性

本文讨论一般非线性随机延迟微分方程Heun方法的数值稳定性,证明了如果问题本身满足零解是均方指数稳定和均方渐近稳定的充分条件,则当方程的漂移项进一步满足一定的条件时,Heun方法是Ms.稳定的,带线性插值的Heun方法是均方指数稳定的和GMS-稳定的理论结果.文末的数值试验进一步验证了所得的相关结论.     

分段连续型随机微分方程指数Euler方法的稳定性

给出了线性分段连续型随机微分方程指数Euler方法的均方指数稳定性.经典的对稳定性理论分析,通常应用的是Lyapunov泛函理论,然而,应用该方程本身的特点和矩阵范数的定义给出了该方程精确解的均方稳定性.以往对于该方程应用隐式Euler方法得到对于任意步长数值解的均方稳定性,而应用显式Euler方法得到了相同的结果.最后,给出实例验证结论的有效性.     

随机延迟微分方程的Milstein方法的非线性均方稳定性

本文针对一般的非线性随机延迟微分方程,证明了当系统理论解满足均方稳定性条件时,则当方程的漂移和扩散项满足一定的条件时,Milstein方法也是均方稳定的.数学实验进一步验证了我们的结论.     

随机延迟微分方程平衡方法的均方收敛性与稳定性

本文讨论求解刚性随机延迟微分方程的平衡方法.证明了随机延迟微分方程平衡方法的均方收敛阶为1/2.给出了线性随机延迟微分方程平衡方法均方稳定的条件.     

A-Stability and Stochastic Mean-Square Stability

This note extends and interprets a result of Saito and Mitsui [SIAM J. Numer. Anal., 33 (1996), pp. 2254–2267] for a method of Milstein. The result concerns mean-square stability on a stochastic differential equation test problem with multiplicative noise. The numerical method reduces to the Theta Method on deterministic problems. Saito and Mitsui showed that the deterministic A-stability property of the Theta Method does not carry through to the mean-square context in general, and gave a condition under which unconditional stability holds. The main purpose of this note is to emphasize that the approach of Saito and Mitsui makes it possible to quantify precisely the point where unconditional stability is lost in terms of the ratio of the drift (deterministic) and diffusion (stochastic) coefficients. This leads to a concept akin to deterministic A()-stability that may be useful in the stability analysis of more general methods. It is also shown that mean-square A-stability is recovered if the Theta Method parameter is increased beyond its normal range to the value 3/2.     

Stochastic Theta Method for a Reflected Stochastic Differential Equation

In this article, a stochastic theta method for a reflected stochastic differential equation is proposed. When the parameter θ = 0, this method coincides with the projection Euler scheme; while when the parameter θ = 1, it is called an implicit projection Euler scheme which is first proposed in this article. Under some conditions, the strong convergence and the A-stability of this numerical scheme are proved.     

Stochastic inertial manifold

A nonlinear stochastic evolution equation in Hilbert space with generalized additive white noise is considered. A concept of stochastic mertial manifold is introduced, defined as a random manifold depending on time, which is finite dimensional, invariant for the dynamic, and attracts exponentially fast all the trajectories as t → ∞. Under the classical spectral gap condition of the deterministic theory, the existence of a stochastic inertial manifold is proved. It is obtained as the solution of a stochastic partial differential equation of degenerate parabolic type, studied by a variant of Bernstein method. A result of existence and uniqueness of a stationary inertial manifold is also proved; the stationary inertial manifold contains the random attractor, introduced in previous works.     

Extrapolation of the Stochastic Theta Numerical Method for Stochastic Differential Equations

ABSTRACT

The stochastic theta method is a family of implicit Euler methods for approximating solutions to Itô stochastic differential equations. It is proved that the weak error for the stochastic theta numerical method is of the correct form to apply Richardson extrapolation. Several computational examples illustrate the improvement in accuracy of the approximations when applying extrapolation.     

Stochastic methods for Dirichlet problems

Using an equivalent expression for solutions of second order Dirichlet problems in terms of Ito type stochastic differential equations, we develop a numerical solution method for Dirichlet boundary value problems. It is possible with this idea to solve for solution values of a partial differential equation at isolated points without having to construct any kind of mesh and without knowing approximations for the solution at any other points. Our method is similar to a recently published approach, but differs primarily in the handling of the boundary. Some numerical examples are presented, applying these techniques to model Laplace and Poisson equations on the unit disk. Visiting Professor, Universidad de Salamanca.     

求解随机微分方程的二级Milstein方法(英文)

In this paper we discuss two-stage Miistein methods for solving Ito stochastic differential equations （SDEs）. Six fully explicit methods （TSM 1 -- TSM 6） are given in this paper. Their order of strong convergence is proved. The stability properties and numerical results show the effectiveness of these methods in the pathwise approximation of Ito SDEs.     

服务系统中位相型分布的随机比较

本文运用应用概率中的随机占优研究位相型(PH)分布的随机比较问题,具体给出在一阶、二阶随机占优下比较两个离散PH分布或两个连续PH分布的充分条件及充分必要条件。研究表明,比较两个离散PH分布可变性的条件与比较两个连续PH分布可变性的条件不同,在二阶随机占优意义下比较两个连续PH分布的条件与均值无关,而比较两个离散PH分布的条件与均值有关。本文的结果可用于研究PH分布的最小变异系数问题和可变性问题,也可用于研究带有PH到达间隔或PH服务的排队系统中到达过程或服务时间可变性对系统队长或等待时间的影响。     

Stochastic programming for nurse assignment

We present a brief overview of four phases of nurse planning. For the last phase, which assigns nurses to patients, a stochastic integer programming model is developed. A Benders’ decomposition approach is proposed to solve this problem, and a greedy algorithm is employed to solve the recourse subproblem. To improve the efficiency of the algorithm, we introduce sets of valid inequalities to strengthen a relaxed master problem. Computational results are provided based upon data from Baylor Regional Medical Center in Grapevine, Texas. Finally, areas of future research are discussed.     

Wick型随机KG方程的精确解

利用白噪声分析、Hermite变换和双曲正切法来研究随机偏微分KleinGordon方程,并在Kondratiev分布空间(S)-1-上分别获得了变系数Klein-Gordon方程和Wick型随机Klein-Gordon方程的精确解和白噪声泛函解.     

Stochastic algorithms with armijo stepsizes for minimization of functions

Stochastic algorithms for optimization problems, where function evaluations are done by Monte Carlo simulations, are presented. At each iteratex _i, they draw a predetermined numbern(i) of sample points from an underlying probability space; based on these sample points, they compute a feasible-descent direction, an Armijo stepsize, and the next iteratex _i+1. For an appropriate optimality function , corresponding to an optimality condition, it is shown that, ifn(i) , then (x _i) 0, whereJ is a set of integers whose upper density is zero. First, convergence is shown for a general algorithm prototype: then, a steepest-descent algorithm for unconstrained problems and a feasible-direction algorithm for problems with inequality constraints are developed. A numerical example is supplied.