建立消费经济学微观数学模型的马尔可夫骨架过程方法
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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。
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.
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。
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.
引文
[1]Markov A.A.Extension of the law of large numbers to dependent quantities inRussian.IZV.FIZ-Matem.Kazan Univ.(2~(nd)ser),1906(15):135-156.
[2]侯振挺.马尔可夫骨架过程-混杂系统模型.长沙:湖南科学技术出版社,2000.
[3]侯振挺.生灭过程.长沙:湖南科学技术出版社,2000.
[4]梅纳德·凯恩斯.就业、利息和货币通论.北京:商务印书馆,1977.
[5]Duesenberry,James S.,Income,saving,and the theory of consumption behavior.Cambridge,Mass.Harvard University Press,1949.
[6]Frideman.M.,A Theory of the Consumption Function[M]Princeton University Press,1957.
[7]Campbell,John Y.,"Consumption,income and interest rates:reinterpreting the time series evidence",in Blanchard,Olivier J.and Stanley Fisher(eds.)NBER Macroeconomics Annual[M],Cambridge,Mass.MIT Press,1989,185-216.
[8]Hall,R.E.,Stochastic implications of the life-cycle permanent income hypotheses:theory and evidence[J],Journal of Political Economy,1978,86,971-987.
[9]吴有昌.中国城乡居民消费函数比较.经济科学,1995(3).
[10]藏旭恒.中国消费函数分析.上海:上海三联书店,上海人民出版社,1994.
[11]李晓西.转轨经济中的消费行为及理论假说.经济科学,1998(4).
[12]程永宏,王长友,曹辉.总消费需求与收入分配的关系.上海经济研究,1999,(7):16-20.
[13]余永定、李军.中国消费函数的理论与验证.中国社会科学,2000(1).
[14]李振明.经济转型与居民消费结构演进.北京:经济科学出版社,2002.
[15]孙凤.中国居民消费行为研究.统计研究,2001(4).
[16]马克思.资本论.北京:人民出版社,1975,第一卷654.
[17]加德纳·阿克利.宏观经济理论.上海:上海译文出版社,1981.240.
[18]尹世杰.消费经济学.北京:高等教育出版社,2003.
[19]伊志宏.消费经济学.北京:中国人民大学出版社,2004.
[20]龚志民.消费经济学前沿.北京:经济科学出版社,2002.
[21]汪同三.宏观经济模型论述.北京:经济管理出版社,1992.
[22]贺菊煌.消费函数研究.北京:社会科学文献出版社,1998.
[23]易丹辉,尹德光.居民消费统计学.北京:中国人民大学出版社,1994.
[24]藏旭恒.消费经济:理论与实证分析.山东大学出版社,1996.
[25]黄亚钧,姜纬.微观经济学教程.上海:复旦大学出版社,1996.63-80.
[26][美]吉利斯等.发展经济学.北京:中国人民大学出版社,1998.
[27]张丽.中国消费函数建模研究.东北财经大学,2002.
[28]Hou Zhenting,Liu Guoxing.Markov Skeleton Processes and their applications.Beijing:Science Press,2005.
[29]Hou Zhenting,Liu Guoxing.Markov Skeleton Process and their applications.School of Manthematics,Central South University.March 2005.
[30]Hou Zhenting,Liu Zaiming,Zou Jiezhong.QNQL-Process:(H,Q)-process and their Application.Chinese Science Bulletin,1997,42(11):881-886.
[31]Hou Zhenting,Liu Zalming,Zou Jiezhong.Markov skeleton processes.Chinese Science Bulletin,1998,43(11):881-889.
[32]侯振挺,郭先平.马尔可夫决策过程.长沙:湖南科学技术出版社,1997.
[33]Hou Zhenting.Markov Skeleton Process and applications to queuing systems.Acta,Mathematic Application Sinica,English Series,2002,18(4):537-552
[34]Hou Zhenting,Yuan Chenggui,Zou Jiezhong et al.Transient distribution of the length of GI/G/n queuing systems.Stochastic Analysis and Applications,2003,21(3):567-592.
[35]叶阿忠.福建省居民收入与消费模型及其预测.福州大学学报(社会科学版),1997(3).
[36]杭斌.经济改革与居民消费行为的渐变一系统变参数模型的应用.山西财经大学学报,1995(6).
[37]杭斌.山西城镇居民消费与收入关系的协整研究.山西财经大学学报,2001(1).
[38]尹清非,近20年来消费函数的新发展[J]1,湘潭大学学报(哲学社会科学版),2004(1).
[39]唐文进.西方消费函数理论的新进展1月.经济学动态,1999(3).
[40]唐文进,段海虹.我国消费与收入的总量分析.消费经济,1995(6).
[41]朱国林.消费理论的最新发展述评.经济学动态,2002(4).
[42]王益民.马尔可夫骨架过程在GI/G/n排队系统中的应用:[博士学位论文],长沙:中南大学,2002.
[43]蒋放鸣.马尔可夫骨架过程及其应用:[博士学位论文],长沙:中南大学,2004.
[44]戴清.马尔可夫骨架过程及其在Frac/G/1排队轮中的应用:[博士学位论文],长沙:中南大学,2004.
[45]黄奇.马尔可夫骨架过程在可靠性理论中的应用:[博士学位论文],长沙:中南大学,2004.
[46]周永卫.数学模型中的马尔可夫骨架过程方法:[硕士学位论文],长沙:中南大学,2005.
[47]范贺花.马尔可夫骨架过程在数学模型中的应用:[硕士学位论文],长沙:中南大学,2005.
[48]姜启源,谢金星,叶俊.数学模型,北京:高等教育出版社,2003.
[49]Ross S M.Stochastic Processes.New York:John Wily & Sons,1983.
[50]Whittaker,E.T.and Watson,G.N.,A Course of Modern Analysis,Cambridge Univ.Press:Cambridge,1952.
[51]严加安.测度论讲义.北京:科学出版社,1988.
[52]钱敏平.随机过程论.北京:科学出版社,1997.
[53]邓永录,梁之舜.随机点过程及其应用.北京:科学出版社,1992.
[54]田乃硕.休假随机服务系统.北京:北京大学出版社,2001.
[55]奇欢.数学模型方法.武汉:华中理工大学出版社,1996.
[56]刘来福,曾文艺.数学模型与数学建模.北京:北京师范大学出版社,2002.
[57]雷公炎.数学模型讲义.北京:北京大学出版社,2004.
[58]科学出版社名词室编,新英汉数学词汇,北京:科学出版社,2002.
[59]田晖.消费经济学.上海:同济大学出版社,2002.68-90.
[60]龚志民.消费经济学前言.北京:经济科学出版社,2002.130-154.
[61]戴维·罗默.高级宏观经济学.北京:商务印书馆,2004.397-436.
[62]梁来存,廖楚晖.湖北省农村居民消费与收入关系的实证分析.统计与决策,2003,(10).
[63]苏明君.辽宁省城镇居民消费与收入的协整研究.东北财经大学学报,2002,(7).
[64]厉以宁.消费经济学.北京:人民出版社,1984.
[2]侯振挺.马尔可夫骨架过程-混杂系统模型.长沙:湖南科学技术出版社,2000.
[3]侯振挺.生灭过程.长沙:湖南科学技术出版社,2000.
[4]梅纳德·凯恩斯.就业、利息和货币通论.北京:商务印书馆,1977.
[5]Duesenberry,James S.,Income,saving,and the theory of consumption behavior.Cambridge,Mass.Harvard University Press,1949.
[6]Frideman.M.,A Theory of the Consumption Function[M]Princeton University Press,1957.
[7]Campbell,John Y.,"Consumption,income and interest rates:reinterpreting the time series evidence",in Blanchard,Olivier J.and Stanley Fisher(eds.)NBER Macroeconomics Annual[M],Cambridge,Mass.MIT Press,1989,185-216.
[8]Hall,R.E.,Stochastic implications of the life-cycle permanent income hypotheses:theory and evidence[J],Journal of Political Economy,1978,86,971-987.
[9]吴有昌.中国城乡居民消费函数比较.经济科学,1995(3).
[10]藏旭恒.中国消费函数分析.上海:上海三联书店,上海人民出版社,1994.
[11]李晓西.转轨经济中的消费行为及理论假说.经济科学,1998(4).
[12]程永宏,王长友,曹辉.总消费需求与收入分配的关系.上海经济研究,1999,(7):16-20.
[13]余永定、李军.中国消费函数的理论与验证.中国社会科学,2000(1).
[14]李振明.经济转型与居民消费结构演进.北京:经济科学出版社,2002.
[15]孙凤.中国居民消费行为研究.统计研究,2001(4).
[16]马克思.资本论.北京:人民出版社,1975,第一卷654.
[17]加德纳·阿克利.宏观经济理论.上海:上海译文出版社,1981.240.
[18]尹世杰.消费经济学.北京:高等教育出版社,2003.
[19]伊志宏.消费经济学.北京:中国人民大学出版社,2004.
[20]龚志民.消费经济学前沿.北京:经济科学出版社,2002.
[21]汪同三.宏观经济模型论述.北京:经济管理出版社,1992.
[22]贺菊煌.消费函数研究.北京:社会科学文献出版社,1998.
[23]易丹辉,尹德光.居民消费统计学.北京:中国人民大学出版社,1994.
[24]藏旭恒.消费经济:理论与实证分析.山东大学出版社,1996.
[25]黄亚钧,姜纬.微观经济学教程.上海:复旦大学出版社,1996.63-80.
[26][美]吉利斯等.发展经济学.北京:中国人民大学出版社,1998.
[27]张丽.中国消费函数建模研究.东北财经大学,2002.
[28]Hou Zhenting,Liu Guoxing.Markov Skeleton Processes and their applications.Beijing:Science Press,2005.
[29]Hou Zhenting,Liu Guoxing.Markov Skeleton Process and their applications.School of Manthematics,Central South University.March 2005.
[30]Hou Zhenting,Liu Zaiming,Zou Jiezhong.QNQL-Process:(H,Q)-process and their Application.Chinese Science Bulletin,1997,42(11):881-886.
[31]Hou Zhenting,Liu Zalming,Zou Jiezhong.Markov skeleton processes.Chinese Science Bulletin,1998,43(11):881-889.
[32]侯振挺,郭先平.马尔可夫决策过程.长沙:湖南科学技术出版社,1997.
[33]Hou Zhenting.Markov Skeleton Process and applications to queuing systems.Acta,Mathematic Application Sinica,English Series,2002,18(4):537-552
[34]Hou Zhenting,Yuan Chenggui,Zou Jiezhong et al.Transient distribution of the length of GI/G/n queuing systems.Stochastic Analysis and Applications,2003,21(3):567-592.
[35]叶阿忠.福建省居民收入与消费模型及其预测.福州大学学报(社会科学版),1997(3).
[36]杭斌.经济改革与居民消费行为的渐变一系统变参数模型的应用.山西财经大学学报,1995(6).
[37]杭斌.山西城镇居民消费与收入关系的协整研究.山西财经大学学报,2001(1).
[38]尹清非,近20年来消费函数的新发展[J]1,湘潭大学学报(哲学社会科学版),2004(1).
[39]唐文进.西方消费函数理论的新进展1月.经济学动态,1999(3).
[40]唐文进,段海虹.我国消费与收入的总量分析.消费经济,1995(6).
[41]朱国林.消费理论的最新发展述评.经济学动态,2002(4).
[42]王益民.马尔可夫骨架过程在GI/G/n排队系统中的应用:[博士学位论文],长沙:中南大学,2002.
[43]蒋放鸣.马尔可夫骨架过程及其应用:[博士学位论文],长沙:中南大学,2004.
[44]戴清.马尔可夫骨架过程及其在Frac/G/1排队轮中的应用:[博士学位论文],长沙:中南大学,2004.
[45]黄奇.马尔可夫骨架过程在可靠性理论中的应用:[博士学位论文],长沙:中南大学,2004.
[46]周永卫.数学模型中的马尔可夫骨架过程方法:[硕士学位论文],长沙:中南大学,2005.
[47]范贺花.马尔可夫骨架过程在数学模型中的应用:[硕士学位论文],长沙:中南大学,2005.
[48]姜启源,谢金星,叶俊.数学模型,北京:高等教育出版社,2003.
[49]Ross S M.Stochastic Processes.New York:John Wily & Sons,1983.
[50]Whittaker,E.T.and Watson,G.N.,A Course of Modern Analysis,Cambridge Univ.Press:Cambridge,1952.
[51]严加安.测度论讲义.北京:科学出版社,1988.
[52]钱敏平.随机过程论.北京:科学出版社,1997.
[53]邓永录,梁之舜.随机点过程及其应用.北京:科学出版社,1992.
[54]田乃硕.休假随机服务系统.北京:北京大学出版社,2001.
[55]奇欢.数学模型方法.武汉:华中理工大学出版社,1996.
[56]刘来福,曾文艺.数学模型与数学建模.北京:北京师范大学出版社,2002.
[57]雷公炎.数学模型讲义.北京:北京大学出版社,2004.
[58]科学出版社名词室编,新英汉数学词汇,北京:科学出版社,2002.
[59]田晖.消费经济学.上海:同济大学出版社,2002.68-90.
[60]龚志民.消费经济学前言.北京:经济科学出版社,2002.130-154.
[61]戴维·罗默.高级宏观经济学.北京:商务印书馆,2004.397-436.
[62]梁来存,廖楚晖.湖北省农村居民消费与收入关系的实证分析.统计与决策,2003,(10).
[63]苏明君.辽宁省城镇居民消费与收入的协整研究.东北财经大学学报,2002,(7).
[64]厉以宁.消费经济学.北京:人民出版社,1984.