建立消费经济学微观数学模型的马尔可夫骨架过程方法

【作者】 曹令秋；

【导师】 侯振挺；

【作者基本信息】 中南大学 ， 概率论与数理统计， 2007， 硕士

【摘要】 马尔可夫骨架过程是一类较为综合的随机过程,它包含了许多已有的随机过程模型,如马尔可夫过程、半马尔可夫过程、逐段决定的马尔可夫过程等一系列的经典的随机过程,具有重要的理论和应用价值。1997年,侯振挺教授等人首次提出马尔可夫骨架过程,并且将其应用于排队论、控制论等领域,成功地解决了排队论的瞬时分布、平稳分布、遍历性等一系列的经典难题,并且提出了许多新问题和新思想。本文主要是利用马尔可夫骨架过程理论来研究消费经济学的微观数学模型,给出了消费系统的持有资金量L(t)的统计规律,研究了消费系统的收入时间间隔和支出时间间隔均服从一般分布时持有资金L(t)的瞬时分布,给出了一种无债消费系统的持有资金量L(t)的瞬时分布及平稳分布。本文的主要结果有:第一、利用马尔可夫骨架过程法列出了在任何时刻t消费系统持有的资金量的瞬时分布及其所满足的线性积分方程,并证明其概率分布是该方程的最小非负解。第二、作为特殊情况给出了一种无债消费系统持有的资金量的瞬时分布及平稳分布。更多还原

【Abstract】 Markov skeleton process is a new type of stochastic process and containing many classical processes such as Markov process, semi-Markov process, piecewise deterministic Markov process. It is very important in theory and application. In 1997, Prof. Hou Zhenting and his colleagues raised this kind of process and applied it to fields in queue theory and cybernetics and solved many difficult queuing theory problems such as transient distribution, stationary distribution and ergodicity successfully.By applying the theory of Markov skeleton process, this paper mainly studies the micro mathematics model of consumer economics, presents the statistical rules of holding the amount of funds in consumption system, studies the transient distribution of holding funds when both income time intervals and expense time intervals are satisfied with regular distribution, and presents a transient distribution and stationary distribution of holding funds in no debt consumption system.In this dissertation, we drew the following conclusions:Firstly, applying the Markov skeleton process method, we can get the transient distribution of holding funds in consumption system in any time and the satisfied linear integral equations .we can also prove that the probability distribution is the minimal nonnegative solution of the equation.Secondly, we present a transient distribution and stationary distribution of holding funds in no debt consumption system as a special case.更多还原