The stochastic ordering of mean-preserving transformations and its applications

The stochastic variability measures the degree of uncertainty for random demand and/or price in various operations problems. Its ordering property under mean-preserving transformation allows us to study the impact of demand/price uncertainty on the optimal decisions and the associated objective values. Based on Chebyshev’s algebraic inequality, we provide a general framework for stochastic variability ordering under any mean-preserving transformation that can be parameterized by a single scalar, and apply it to a broad class of specific transformations, including the widely used mean-preserving affine transformation, truncation, and capping. The application to mean-preserving affine transformation rectifies an incorrect proof of an important result in the inventory literature, which has gone unnoticed for more than two decades. The application to mean-preserving truncation addresses inventory strategies in decentralized supply chains, and the application to mean-preserving capping sheds light on using option contracts for procurement risk management.     

Normal approximations for discrete-time occupancy processes

We study normal approximations for a class of discrete-time occupancy processes, namely, Markov chains with transition kernels of product Bernoulli form. This class encompasses numerous models which appear in the complex networks literature, including stochastic patch occupancy models in ecology, network models in epidemiology, and a variety of dynamic random graph models. Bounds on the rate of convergence for a central limit theorem are obtained using Stein’s method and moment inequalities on the deviation from an analogous deterministic model. As a consequence, our work also implies a uniform law of large numbers for a subclass of these processes.     

大数据背景下水厂自动投矾模型研究

根据某市自来水有限责任公司第二水厂的历史矾耗数据，建立矾耗流量关于原水浊度、温度等的动态矾耗模型. 通过对数据进行处理得到10900个合格且净水效果高效的数据，将筛选出的数据分为训练样本集和测试样本集. 在回归拟合中，通过拟合R2的大小将原水浊度划分为“低浊”“中浊”“高浊”3个区间，利用泰勒展开公式的非线性变量代换分别对3个区间建立不同的多项式回归模型，得到预测正确率约为72%，总的矾耗流量值约减少了9.6%的结果；在随机森林模型中，使用10900个合格数据，利用训练样本集，以“原水浊度”“pH值”“原水流量”和“水温”为输入变量，建立包含2000棵决策树的随机森林模型，得到预测正确率约为44. 21%，总的矾耗流量值增加了0.04%的结果. 从模型对合格数据的拟合优度看，随机森林模型比非线性回归模型效果更好；在平均绝对误差、平均绝对偏差百分比等评价指标上，前者均优于后者；但从历史数据检验的结果，模型的可解读性，模型的操作难度和推广角度看，分段二元非线性回归模型的优势更为突出.     

个体投资治理污染的随机增长模型

本文研究了个体投资治理污染的随机增长模型.利用随机最优化的方法,得出了随机扰动、个体环保投资及环保技术对福利和经济增长的影响.对我国制定环保政策具有一定的积极作用.     

Stochastic fractional differential equations: Modeling,method and analysis

By introducing a concept of dynamic process operating under multi-time scales in sciences and engineering, a mathematical model described by a system of multi-time scale stochastic differential equations is formulated. The classical Picard–Lindelöf successive approximations scheme is applied to the model validation problem, namely, existence and uniqueness of solution process. Naturally, this leads to the problem of finding closed form solutions of both linear and nonlinear multi-time scale stochastic differential equations of Itô–Doob type. Finally, to illustrate the scope of ideas and presented results, multi-time scale stochastic models for ecological and epidemiological processes in population dynamic are outlined.     

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。     

Dynamics of a general non-autonomous stochastic Lotka-Volterra model with delays

In this paper, a general non-autonomous n-species Lotka-Volterra model with delays and stochastic perturbation is investigated. For this model, sufficient conditions for extinction, non-persistence in the mean, weak persistence and stochastic permanence are established. The influences of the stochastic noises to the properties of the stochastic model are discussed. The property permanence for the model is preserved with the sufficiently small noise and sufficiently large noise may cause extinction of the model. The critical value between weak persistence and extinction is obtained. Finally, numerical simulations are given to support the theoretical analysis results.     

Strong convergence rate of principle of averaging for jump-diffusion processes

We study jump-diffusion processes with two well-separated time scales. It is proved that the rate of strong convergence to the averaged effective dynamics is of order O(ɛ ^1/2), where ɛ ≪ 1 is the parameter measuring the disparity of the time scales in the system. The convergence rate is shown to be optimal through examples. The result sheds light on the designing of efficient numerical methods for multiscale stochastic dynamics.     

LONG-TERM DYNAMIC ANALYSIS OF ENDANGERED SPECIES WITH STAGE-STRUCTURE AND MIGRATIONS IN POLLUTED ENVIRONMENTS

We propose a stochastic stage-structured single-species model with migrations and hunting within a polluted environment, where the species is separated into two groups: the immature and the mature, which migrates from one patch to another with different migration rates. By constructing a Lyapunov function, together with stochastic analysis approach, the stochastic single-species model admits a unique global positive solution. We then utilize the comparison theorem of stochastic differential equations to investigate the extinction and persistence of solution to stochastic single-species model. The main results indicate that the species densities all depend on the intensities of random perturbations within both patches. As a consequence, we further provide several strategies for protecting endangered species within protected and unprotected patches.     

GI(1)+GI(2)+…+GI(N)/M/1排队模型

文献[1]引入了一类具有广泛应用前景的随机过程--Markov骨架过程.本文借助这类随机过程的方法研究了GI(1)+GI(2)+…+GI(N)/M/1排队模型,求出了此模型到达过程、等待时间及队长的概率分布.     

随机过程增量不等式的推广

本文将推广在［３］中由Ｅ．Ｃｓａｋｉ及Ｍ．Ｃｓｏｒｇｏ所引入的关于随机过程不等式，并把它应用到某些随机过程中，从而得到这些随机过程的一些极限定理．     

Models of stochastic geometry — A survey

This paper discusses some models of stochastic geometry which are of potential interest for operations research. These are the Boolean model, a certain model for random compact sets and marked point processes. The Boolean model is a generalization of the well-known queueing systemM/G/. The random compact set model may be useful for modelling spatial spreading processes such as fires, cancers or holes in the Earth's surface. Marked point processes are used here as models of forests and used for a statistical study of the spatial distribution of damaged trees.Extended version of an Invited Lecture on the 16th Symposium for OR in Hamburg 1992.     

THE EFFECTS OF REDUCING MATING LIKELIHOOD ON POPULATION VIABILITY

Abstract The success a species may have invading a patch previously unoccupied is of considerable interest for pest managers and conservation ecologists. The purpose here is to present a mechanistic approach to analyze reproductive Allee effects appearing through the failure in the process of fertilization in a two‐sex population and observe how the survival in an invaded patch is affected. This is in contrast to the usually employed stochastic models with a deterministic skeleton that describe the presence of Allee effects. A Poisson–Ricker model, which includes stochastic demography and sex determination with females classified as successfully fertilized or not fertilized, is used. Numerical approximations to the probabilities of extinction and the mean time to extinction are presented, for fixed parameter values, suggesting how stochasticity in the mating process combined with random fluctuations in the male and female densities, at each generation, contribute to the risk of extinction of a population which started an invasion at a low density.     

Parametric inference for discrete observations of diffusion processes with mixed effects

Stochastic differential equations with mixed effects provide means to model intra-individual and inter-individual variability in repeated experiments leading to longitudinal data. We consider N i.i.d. stochastic processes defined by a stochastic differential equation with linear mixed effects which are discretely observed. We study a parametric framework with distributions leading to explicit approximate likelihood functions and investigate the asymptotic behavior of estimators under the asymptotic framework : the number N of individuals (trajectories) and the number n of observations per individual tend to infinity within a fixed time interval. The estimation method is assessed on simulated data for various models.     

Multiscale analysis and reconstruction of time series of stochastic cascade processes

We propose a new method to generate synthetical time series of hierarchical stochastic processes. Based on the statistics of n–scale joint PDFs, the stochastic properties of a time series are modeled simultaneously on many scales. The application to a data set of turbulent velocities is demonstrated, showing the ability of the approach to reproduce the correct statistics of the original time series on all considered scales. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the weak convergence of Wright-Fisher model

We prove that the sequence of stochastic processes obtained from Wright-Fisher models by transforming the time scales and state spaces in the usual way converges weakly to a diffusion process on the time interval [0,∞). Convergence of fixation probabilities and fixation time distributions are obtained as corollaries. These results extend a theorem of Watterson, who proved convergence in distribution to a diffusion at any given single time point for these processes.     

Stochastic modeling of phytoplankton allelopathy

The study of dynamic interactions between two competing phytoplankton species in the presence of toxic substances is an active field of research due to the global increase of harmful phytoplankton blooms. Ordinary differential equation models for two competing phytoplankton species, when one or both the species liberate toxic substances, are unable to capture the oscillatory and highly variable growth of phytoplankton populations. The deterministic formulation never predicts the sudden localized extinction of certain species. These obstacles of mathematical modeling can be overcome if we include stochastic variability in our modeling approach. In this investigation, we construct stochastic models of allelopathic interactions between two competing phytoplankton species as a continuous time Markov chain model as well as an Itô stochastic differential equation model. Approximate extinction probabilities for both species are obtained analytically for the continuous time Markov chain model. Analytical estimates are validated with the help of numerical simulations.     

Stability of stochastic SIRS epidemic models with saturated incidence rates and delay

In this article, we consider stochastic susceptible-infected-removed-susceptible (SIRS) epidemic models with saturated incidence rates and delay. We investigate the stochastic stability in probability of the disease-free and endemic equilibria for the stochastic dynamic model with variability in the natural death rate, and the stochastic stability in probability of the endemic equilibrium for the dynamic model when the variability in the environment is proportional to a deviation between the state of the system and the endemic equilibrium. The numerical experiments are provided to support our theoretical results.     

Stochasticity in the image greenhouse model

The potential impacts of climate change are a major issue in the Greenhouse debate. Various models and particularly IMAGE (Integrated Model to Assess the Greenhouse Effect), are being used by the IPCC to investigate climate change. The IMAGE model has been reduced to a system of differential equations and incorporates various initial conditions and model parameters. These initial conditions and parameter values are not known precisely and are subject to variability of various forms. This paper briefly describes the mathematical form of IMAGE and then investigates the stochastic properties of the model. Particular attention is paid to the propagation and amplification of assumed distributions for the initial conditions and certain key physical parameters. The IMAGE model appears to be quite robust with respect to these stochastic characteristics.     

需求可变性降低对均值CVaR约束销售努力决策库存系统的影响

本文运用Levy提出的变换研究需求可变性降低对风险偏好零售商的库存决策、销售努力决策和期望效用的影响,用均值CVaR刻画零售商的风险偏好特性,它包括风险厌恶、风险追求,也具有损失规避的特性。首先,运用该变换定量刻画需求可变性的降低,证明该变换蕴含经典随机占优中的割准则序和二阶随机占优等。其次,给出系统的最优订货量、最优期望效用和最优销售努力水平,得到它们关于风险偏好系数的单调性,并给出降低需求可变性对期望效用的影响。第三,针对风险中性、风险厌恶(最大化CVaR)和风险追求(最小化CVaR)这三种特殊情况得到相应的结果,并给出企业在库存决策和促销决策的管理启示。最后,通过数值例子验证了得到的研究结果并给出相应的管理启示。