奖惩系统的数学建模与稳态分析

奖惩系统(Bonus- Malus Systems)是世界各国机动车辆险中广泛采用的一种经验费率厘定机制.本文在最一般的框架下,以有限齐次马尔科夫链对奖惩系统建模,证明了任一奖惩系统皆存在唯一的平稳分布,此外,给出了求解这一平稳分布的一般算法,并揭示了此平稳分布的结构.特别地,针对两种具有特定转移法则的奖惩系统,还给出了它们平稳分布的简明显式表示.     

具有标准发生率的随机SIR传染病模型的建立及平稳分布与疾病灭绝研究

研究了一类具有标准发生率以及考虑随机扰动与系统变量成正比的随机SIR传染病模型.首先,对于任意的正的初值,系统存在唯一的全局正解以及通过构造合适的随机李雅普诺夫函数,得到了模型遍历平稳分布存在的充分条件.其次,给出了疾病灭绝的充分条件,并与模型遍历平稳分布存在的充分条件作对比,得出了在特定条件下随机SIR模型的阈值.最后通过数值模拟验证了结果的正确性.     

一类随机系统平稳分布的存在性与唯一性

本文研究了一类随机系统平稳分布存在唯一性问题.利用耦合方法,给出了一个充分条件,验证该条件只需计算一些耦合,具较强的可操作性.     

一类带跳平均场泛函随机微分方程的平稳分布

平均场方法是用来研究复杂系统的重要工具,广泛应用于各个研究领域.自Buckdahn等人提出平均场倒向随机微分方程以来,平均场方法在随机分析的应用受到了越来越多学者的关注.本文研究一类L′evy过程驱动的平均场泛函随机微分方程,基于依分布收敛的思想,对其平稳分布进行分析,得到平稳分布存在唯一性的充分条件.     

马氏链平稳分布存在与唯一性的简洁证明与计算

本文考虑可数状态离散时间齐次马氏链平稳分布的存在与唯一性.放弃以往大多数文献中要求马氏链是不可约,正常返且非周期(即遍历)的条件,本文仅需要马氏链是不可约和正常返的(但可能是周期的,因而可能是非遍历的).在此较弱的条件下,本文不仅给出了平稳分布存在与唯一性的简洁证明,而且还给出了平稳分布的计算方法.     

随机环境中马氏链的平稳分布(英文)

本文引入了随机环境中马氏链平稳分布的概念. 在合适的条件下, 给出了随机环境中马氏链的平稳分布存在的一些充分条件. 特别地, 讨论了Cogburn链的平稳分布存在性问题. 同时, 构造了一个随机环境中马氏链的例子, 它的平稳分布是存在的.     

NCD系统的数学建模与稳态分析

本文以有限齐次马尔科夫链对NCD（无索赔折扣）系统建模，严谨地证明了，任一NCD系统皆存在唯一的平稳分布。此外，给出了求解这一平稳分布的一般算法，并揭示了此平稳分布的结构。特别地，对两类给定的折扣类转移法则，还给出了平稳分布的显式。     

不动点与Markov过程的稳定性

运用不动点方法与耦合技巧得到一般状态空间上Markov过程平稳分布存在唯一性和KRW概率距离下的稳定性判据.作为应用,讨论了扩散过程的稳定性.     

对一类离散时间休假排队平稳指标的数值计算与渐近分析

本文研究休假时间服从T-IPH分布的Geo/Geo/1休假排队,其中T-IPH分布是由可数状态吸收生灭链定义的离散时间无限位相分布.对多重休假和单重休假两种情形,基于系统平稳方程和复分析方法,首先得到了排队系统平稳队长和平稳逗留时间的概率母函数(PGF);其次,通过对PGF分析,进一步得到了平稳附加队长和附加逗留时间分...     

GI/G/1排队系统的几种收敛速度

对随机模型,可以从不同角度研究其稳定性,一种是研究其转移概率函数趋向于平稳分布的速度,即各种遍历性;另一种是研究平稳分布的尾部衰减速度.本文从这两个方面着手,找它们之间的关系,对GI/G/1排队系统,给出等待时间列几何遍历、平稳分布轻尾与服务时间分布轻尾三者等价,l-遍历、平稳分布的尾部(l-1)-阶衰减与服务时间分布的尾部l-阶衰减三者等价,最后证明出等待时间列不是强遍历.     

Existence of the optimal measurable coupling and ergodicity for Markov processes

The existence theorem of the optimal measurable coupling of two probability kernels on a complete separable metric measurable space is proved. Then by this theorem, a general ergodicity theorem for Markov processes is obtained. And as an immediate application to particle systems the uniqueness theorem of the stationary distribution is supplemented, i.e. the uniqueness theorem also implies the existence of the stationary distribution.     

Burgers equation with random boundary conditions

We prove an existence and uniqueness theorem for stationary solutions of the inviscid Burgers equation on a segment with random boundary conditions. We also prove exponential convergence to the stationary distribution.

     

Dynamics of a Stochastic Predator–Prey Model with Stage Structure for Predator and Holling Type II Functional Response

In this paper, we develop and study a stochastic predator–prey model with stage structure for predator and Holling type II functional response. First of all, by constructing a suitable stochastic Lyapunov function, we establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the model. Then, we obtain sufficient conditions for extinction of the predator populations in two cases, that is, the first case is that the prey population survival and the predator populations extinction; the second case is that all the prey and predator populations extinction. The existence of a stationary distribution implies stochastic weak stability. Numerical simulations are carried out to demonstrate the analytical results.     

Stationary distribution and extinction of a stochastic nutrient-phytoplankton-zooplankton model with cell size

In this paper, we study a stochastic nutrient-phytoplankton-zooplankton model with cell size that represents the interaction between internal mechanism of species and external environment. We first investigate the existence and uniqueness of the global positive solution with positive initial values. Then we construct sufficient conditions for the existence of an ergodic stationary distribution of positive solution. Once more, we find that large noise intensities cause the extinctions of phytoplankton and zooplankton. Finally, numerical simulations are given to verify the correctness of theoretical results.     

The threshold of a stochastic Susceptible–Infective epidemic model under regime switching

In this paper, we consider a stochastic Susceptible–Infective (SI) epidemic model under regime switching. Firstly, by constructing suitable Lyapunov functions, we establish sufficient criteria for the existence and uniqueness of an ergodic stationary distribution. Then we obtain the threshold which guarantees the extinction and the existence of the stationary distribution of the epidemic. Finally, some numerical simulations are introduced to illustrate our main results.     

Convergence conditions for Gummel's method

Four data-smallness conditions that guarantee existence and uniqueness for solutions of stationary systems of equations in the physics of semiconductors are derived. Gummel's method converges under these conditions.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 185, pp. 146–159, 1990.     

Quasi-asymptotic contractions,set-valued dynamic systems,uniqueness of endpoints and generalized pseudodistances in uniform spaces

For set-valued dynamic systems in uniform spaces we introduce the concept of quasi-asymptotic contractions with respect to some generalized pseudodistances, describe a method which we use to establish general conditions guaranteeing the existence and uniqueness of endpoints (stationary points) of these contractions and exhibit conditions such that for each starting point each generalized sequence of iterations (in particular, each dynamic process) converges and the limit is an endpoint. The definition, result, ideas and techniques are new for set-valued dynamic systems in uniform, locally convex and metric spaces and even for single-valued maps.     

Stationary distribution of the stochastic theta method for nonlinear stochastic differential equations

Numerical Algorithms - The existence and uniqueness of the stationary distribution of the numerical solution generated by the stochastic theta method are studied. When the parameter theta takes...     

Dynamical behavior of a stochastic predator-prey model with stage structure for prey

Abstract

In the present paper, we focus on a stochastic predator-prey model with stage structure for prey. Firstly, by using the stochastic Lyapunov function method, we obtain sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the model. Then we establish sufficient conditions for extinction of the predator population in two cases. Some examples and numerical simulations are carried out to validate our analytical findings.