马尔可夫骨架过程的两个应用

【作者】 李占光；

【导师】 侯振挺；

【作者基本信息】 中南大学 ， 概率论与数理统计， 2006， 硕士

【摘要】 本文主要是利用马尔可夫骨架过程理论来研究单一奶牛群个体数量增长模型和家庭收支模型。对于单一奶牛群个体数量增长模型，与前人的工作相比，本文所研究的模型中各个参数均是服从为一般的分布。而前人所研究的主要是对单种群增长模型进行分析，数据拟合，从而建立起一个常微分模型方程组，或者一阶、二阶偏微分方程组，通过解微分方程组来研究单种群的增长规律，建立的一般为马尔萨斯模型和有阻滞的Logistic增长模型。对于家庭收支模型，本文所研究的模型是随机环境流体模型的一个推广，而前人对家庭收支模型的研究较少。在本文中，研究了家庭收支模型中家庭帐户上资金的瞬时分布。同时，改进了家庭收支模型，使之与实际情况较为贴近。针对以上两种模型，本文利用的是侯振挺、邹捷中、刘再明教授等人于1997年首次提出的，并在最近的2005年加以补充完善的马尔可夫骨架过程理论来研究模型中某一些状态的瞬时分布。本文的研究结果主要有：第一，利用了马尔可夫骨架过程法列出了对单种群增长模型中种群个体数{L（t），（?）（t），θ₁（t），θ₂（t），…，θ_L（t）（t），（?）（t）}的瞬时分布所满足的方程组，并对概率分布是某一方程的最小非负解作了证明。第二，利用了马尔可夫骨架过程的向后理论和骨架时序列的逐段决定的骨架过程理论对家庭收支模型中家庭帐户中资金数量的瞬时分布给出了解的存在过程，并对其概率分布是某一方程的最小非负解作了证明。更多还原

【Abstract】 This text mainly applies the process theories of the Markov skeleton process to study the growth model of the individual amount of single cow herd of cattle and family receipt and expenditure model.For the growth model of the individual amount of single cow herd of cattle, compared with the work made in the past each parameter in this studied model is all obedience to the general distribution. While what the past study is mainly to analyse the increase model of single herd kinds to increase draw up the data to match, so as to build one thus the differential calculus model square distance set, perhaps a rank, two ranks partial to the differential calculus square distance set. By solving the differential calculus square distance set, the growth regulation of single herd is studied, and the Malthus model and the growth model of the Logistic of resistance are built up.For family receipt and expenditure model, the model studied is this text’s an expansion of a random environment fuiid model a random environment fluid is an expansion of the model, but the past studies made few researches on family receipt and expenditure model. The text explores the moment distribution of family bank account in the family receipt and expenditure model. At the same time, the family receipt and expenditure model is improved to meet the actual situation. As for above two kinds of models, this text depends on what Hou.Zhenting, Zou.Jiezhong, Liu.Zaiming professor etc. Put forward for the first time in 1997 and the perfect Markov skeleton process theories which was improved in 2005 to study the moment distribution under some condition in the model.The results of the study mainly include:First, in terms of Markov skeleton process method, it lists the square distance set that number{L（t）,θ{t),θ₁（t）,θ₂ （t）,…,θ_L（t）（t）,θ（t）} satisfy when ina moment distribute in the growth model to the single herd, and to prove that all rate’s distributing is a some least of one square distance to take a solution.Second, in terms of Markov skeleton process backward theories and the theory that framework cycle time row pursue a framework process theories of decision the current bank account in the model to the family receipt and expenditure to distribute the existence process of understanding to in a moment in the funds amount, and prove it’s all rate’s distributing is a some least of one square distance to take a solution.更多还原