随机动力系统在投资组合中的应用
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摘要
当今社会已经进入经济竞争的时代，社会发展以及人类进步的要求促使人们对金融经济研究投入越来越多的精力。几十年来，人们不断的发展新的理论方法，希望能够推动数理金融的发展与创新。本文利用随机动力系统理论研究了大家关心的某些经济问题尤其是资产组合问题。
本论文研究的创新点主要表现在以下两大方面：
第一部分是在Klaus的文章“Evolution of Portfolio Rules in Incomplete Market”基础上完成的。Klaus使用随机动力系统理论建立了关于市场份额与投资份额的模型，并得到了将来时刻最优投资策略的显式表达式。这一理论的经济意义是选择最优投资策略的投资者将是市场选择过程唯一的幸存者，将拥有整个市场财富。我们想借鉴Klaus的理论，将其应用到中国的金融市场，但这个结论是否适用于中国市场是个有争议的问题。因此第一部分的工作我们主要集中在实证研究方面。方法是以中国股票市场上2000-2001年的原始股票数据为材料，选择几种具有代表性的股票日收盘价作为基础数据，代入Klaus得到的三个中心计算公式中。本文分别以两种资产、五种资产、十种资产对五位投资者以及十位投资者为例，得到了不同情形下每位投资者在150天内的财富演变情况，分析了每种情况中图形所代表的经济含义。最后我们得出结论：Klaus的理论同样适用于中国的股票市场。
第二部分主要是理论方面的工作。受[1]的启发，我们重新考虑了不完全市场中的投资份额与市场份额。在这部分中允许市场的总财富在整个投资过程中发生改变，同时允许投资者可以撤离部分资金。这弥补Klaus文中提到的条件假设的不足。同样应用随机动力系统理论，我们得到一个新的随机方程模型。利用天数定律以及调和性质，得到了市场总财富发生改变情况下市场份额将来时刻的不动点。可以证明，这个不动点是稳定的。本文最后对此不动点的经济含义给出简单了的说明。
Nowadays, society has entered into economic competition time, the development of society and the requirement for human progress induce people to devote more and more energy to finance and economics research. For decades of development, people increasingly look for better theoretical systems to improve finance development and innovation. Some economic problems especially portfolio are studied in this thesis by use of random dynamical system theory.
This thesis deals with the following two creative topics :
The first part is finished on the base of Klaus' working paper "Evolution of Portfolio Rules in Incomplete Market". Klaus utilize random dynamical system theory to set up a model for market share and invest share. Moreover an explicit formula for the best investment strategy in the future time is derived. The significance this result brought is the investor who selects this strategy will be the only survivor and dominate total market wealth. For this reason, we want to own this result and carry it out on China financial market. However whether this theory adapt to China market is a susceptible problem. Therefore we pay much attention to confirmation study in the first part. The technique we proposed is to take the original stock data on China stock market from year 2000 to 2001 as material, and select final data per day of several representational stocks as fundamental data, then put data into that three important formula. This thesis taking two assets, five assets, ten assets and five investors, ten investors as example, obtains wealth evolution within 150 days. At the same time financial significance involved in the graph is also analyzed in detail. At last we come to the conclusion: Klaus theory in shape with China stock market.
The second part focuses on the theoretical task. Being inspirited by [1], invest share and market share in the incomplete market are considered once more. In this sector market wealth is admitted to change and every investor can withdraw a part of his wealth during investment, which are reverse to those hypotheses of Klaus. We
We place our study within the theory of random dynamical system, as a result we get a new random equation model. With the law of large number and temper property, a fixed point of market share in the future time can be found when total wealth of market is changeable. It is proved that this fixed point is stable. At the end of this thesis, a simple explanation for economic significance of it is provided.
本论文研究的创新点主要表现在以下两大方面：
第一部分是在Klaus的文章“Evolution of Portfolio Rules in Incomplete Market”基础上完成的。Klaus使用随机动力系统理论建立了关于市场份额与投资份额的模型，并得到了将来时刻最优投资策略的显式表达式。这一理论的经济意义是选择最优投资策略的投资者将是市场选择过程唯一的幸存者，将拥有整个市场财富。我们想借鉴Klaus的理论，将其应用到中国的金融市场，但这个结论是否适用于中国市场是个有争议的问题。因此第一部分的工作我们主要集中在实证研究方面。方法是以中国股票市场上2000-2001年的原始股票数据为材料，选择几种具有代表性的股票日收盘价作为基础数据，代入Klaus得到的三个中心计算公式中。本文分别以两种资产、五种资产、十种资产对五位投资者以及十位投资者为例，得到了不同情形下每位投资者在150天内的财富演变情况，分析了每种情况中图形所代表的经济含义。最后我们得出结论：Klaus的理论同样适用于中国的股票市场。
第二部分主要是理论方面的工作。受[1]的启发，我们重新考虑了不完全市场中的投资份额与市场份额。在这部分中允许市场的总财富在整个投资过程中发生改变，同时允许投资者可以撤离部分资金。这弥补Klaus文中提到的条件假设的不足。同样应用随机动力系统理论，我们得到一个新的随机方程模型。利用天数定律以及调和性质，得到了市场总财富发生改变情况下市场份额将来时刻的不动点。可以证明，这个不动点是稳定的。本文最后对此不动点的经济含义给出简单了的说明。
Nowadays, society has entered into economic competition time, the development of society and the requirement for human progress induce people to devote more and more energy to finance and economics research. For decades of development, people increasingly look for better theoretical systems to improve finance development and innovation. Some economic problems especially portfolio are studied in this thesis by use of random dynamical system theory.
This thesis deals with the following two creative topics :
The first part is finished on the base of Klaus' working paper "Evolution of Portfolio Rules in Incomplete Market". Klaus utilize random dynamical system theory to set up a model for market share and invest share. Moreover an explicit formula for the best investment strategy in the future time is derived. The significance this result brought is the investor who selects this strategy will be the only survivor and dominate total market wealth. For this reason, we want to own this result and carry it out on China financial market. However whether this theory adapt to China market is a susceptible problem. Therefore we pay much attention to confirmation study in the first part. The technique we proposed is to take the original stock data on China stock market from year 2000 to 2001 as material, and select final data per day of several representational stocks as fundamental data, then put data into that three important formula. This thesis taking two assets, five assets, ten assets and five investors, ten investors as example, obtains wealth evolution within 150 days. At the same time financial significance involved in the graph is also analyzed in detail. At last we come to the conclusion: Klaus theory in shape with China stock market.
The second part focuses on the theoretical task. Being inspirited by [1], invest share and market share in the incomplete market are considered once more. In this sector market wealth is admitted to change and every investor can withdraw a part of his wealth during investment, which are reverse to those hypotheses of Klaus. We
We place our study within the theory of random dynamical system, as a result we get a new random equation model. With the law of large number and temper property, a fixed point of market share in the future time can be found when total wealth of market is changeable. It is proved that this fixed point is stable. At the end of this thesis, a simple explanation for economic significance of it is provided.
引文
[1] .Thorsten Hens and Schenk-Hoppe,K.R.. An evolutionary portfolio in the incomplete market. Institute for empirical research in economics, 2001．
[2] .Ludwig Anold. Random dynamical system. Springer-Verlag Berlin Heidelberg New York, 1998．
[3] .Igor Evstigneev, Thorsten Hens and Schenk-Hoppe,K.R.. Market selection of financial trading strategies: global stability. Institute for empirical research in economics, 2001．
[4] .Schenk-Hoppe,K.R., Sample-path stability of non-Stationary dynamic economic systems. Institute for empirical research in economics, 2001．
[5] .Blume and Easley. Evolution and market behavior. Journal of economic theory, 1992．
[6] .Cover. An algorithm for maximizing expected log-investment return. IEEE transformation information theory, 1984．
[7] .Breiman, Optimal gambling system for favorable games. Fourth Berkeley symposium on mathematical statistics and probability, 1961．
[8] .Hakansson,N. Optimal investment and consumption strategies under risk for a class of utility functions. Econometrics, 1970,38(4) ,587-607．
[9] .Karatzas,I.and S.Shreve. Methods of mathematical finance. Springer-Verlag, New York, 1998．
[10] .chenk-Hoppe,K.R. Random dynamical systems in economics. Stochastics and dynamics, 2001,1(1) ,63-83．
[11] .Sciubba,E. The evolution of portfolio rules and the capital asset pricing model. Working paper, University of Cambridge, mimeo,1998．
[12] .Thorp E. Portfolio choice and the Kelly criterion. In stochastic models in finance, 1971,4(4) ,599-619．
[13] .Williams J.B. The theory of value. Harvard University press, MA,1938．
[14]. Markowitz H.M. Portfolio selection: efficient diversification of investment. Wiley, New York,NY, 1959.
[15]. Huang Jiangguo, Ye zhangxing. An adaptive algorithm for log-optimal portfolio and its theoretical analysis. 应用概率统计，2001,17(1),88—98.
[16]．唐小我．组合证券投资决策的计算方法．管理工程学报，1990，3，45—48．
[17]．荣喜民，张世英．组合证券资产选择模糊最优化模型研究．系统工程理论与实践，1998，8，26-32．
[18]．Zvi bodie，Alex Kane，Alan J．Msrcus．朱宝宪，吴洪，赵冬青译．投资学．机械工业出版社，北京，2000．
[19]．候为波．现代证券组合选择理论研究．西安交通大学博士学位论文，1999．
[20]．叶中行，林建忠．数理金融—资产定价与金融决策理论．科学出版社，北京，1998．
[21]．Hull．J．张陶伟译．期权、期货和其它衍生产品．华夏出版社，北京，2000．
[22]．候为波，徐成贤．证券组合M—V有效边缘动态分析，系统工程学报，2000，15(1)，26-31．
[23]．马永开，唐小我．限制性卖空条件下β值证券组合投资模型．系统工程学报，2001，16(2)，81—87．
[24]．谭浩强．C程序设计．清华大学出版社，北京，1995．
[25]．肖庆宪，郑祖康．股票价格过程方差函数的统计推断．应用概率统计2000，16(2)，182-190．
[26]．杨昭军，邹捷中．一种亚式期权的套期保值策略．应用数学学报，2001，24(1)，56-60．
[27]．薛红，聂赞坎．Multi-dimensional black-sholes model with non-constant coefficients and dividend．应用数学，2000，13(3)，133-138．
[28]．刘忠．SV模型下的期权定价和风险计量．应用概率统计，2000，16(4)，365-370．
[29]．候为波，徐成贤．证券组合投资决策的β模型．应用数学，2000，13(3)，25-30．
[30]．曾勇，唐小我，郑维敏．基金分离定理的进一步研究．系统工程学报，2001，16(3)，187-191．
[31]．荣喜民，张世英．再保险定价的研究．系统工程学报，2001，16(6)，471-474．
[32]．何基报，茆诗松．对数正态分布场合无失效的Bayes验证试方案．应用概率统计，2000，16(3)，239-248．
[33]．林金官．心理状态数的Bayes估计．数理统计与管理，2001，20(2)，31-35．
[34]．俞迎达．Black-scholes期权定价模型的简化推导．数学的实践与认识，2001，31(6)，756-758．
[35]．Anold．V．I．Ordinary differential equations．沈家骐，周宝熙，卢亭鹤译．科学出版社，北京，2001．
[36]．盛昭瀚，马军海．非线性动力系统分析引论．科学出版社，北京，2001．
[37]．中国证券业协会．证券投资分析．上海财经大学出版社，上海，2002．
[38]．中国证券业协会．证券市场基础知识．上海财经大学出版社，上海，2002．
[39]．钱敏平，龚光鲁．随机过程论，北京大学出版社，北京，1997．
[40]．Sharpe．W．F．Portfolio theory and capiml markets．霍小虎，奚敬泽，王继德，郑启鸣译．中国经济出版社，北京，1992．
[2] .Ludwig Anold. Random dynamical system. Springer-Verlag Berlin Heidelberg New York, 1998．
[3] .Igor Evstigneev, Thorsten Hens and Schenk-Hoppe,K.R.. Market selection of financial trading strategies: global stability. Institute for empirical research in economics, 2001．
[4] .Schenk-Hoppe,K.R., Sample-path stability of non-Stationary dynamic economic systems. Institute for empirical research in economics, 2001．
[5] .Blume and Easley. Evolution and market behavior. Journal of economic theory, 1992．
[6] .Cover. An algorithm for maximizing expected log-investment return. IEEE transformation information theory, 1984．
[7] .Breiman, Optimal gambling system for favorable games. Fourth Berkeley symposium on mathematical statistics and probability, 1961．
[8] .Hakansson,N. Optimal investment and consumption strategies under risk for a class of utility functions. Econometrics, 1970,38(4) ,587-607．
[9] .Karatzas,I.and S.Shreve. Methods of mathematical finance. Springer-Verlag, New York, 1998．
[10] .chenk-Hoppe,K.R. Random dynamical systems in economics. Stochastics and dynamics, 2001,1(1) ,63-83．
[11] .Sciubba,E. The evolution of portfolio rules and the capital asset pricing model. Working paper, University of Cambridge, mimeo,1998．
[12] .Thorp E. Portfolio choice and the Kelly criterion. In stochastic models in finance, 1971,4(4) ,599-619．
[13] .Williams J.B. The theory of value. Harvard University press, MA,1938．
[14]. Markowitz H.M. Portfolio selection: efficient diversification of investment. Wiley, New York,NY, 1959.
[15]. Huang Jiangguo, Ye zhangxing. An adaptive algorithm for log-optimal portfolio and its theoretical analysis. 应用概率统计，2001,17(1),88—98.
[16]．唐小我．组合证券投资决策的计算方法．管理工程学报，1990，3，45—48．
[17]．荣喜民，张世英．组合证券资产选择模糊最优化模型研究．系统工程理论与实践，1998，8，26-32．
[18]．Zvi bodie，Alex Kane，Alan J．Msrcus．朱宝宪，吴洪，赵冬青译．投资学．机械工业出版社，北京，2000．
[19]．候为波．现代证券组合选择理论研究．西安交通大学博士学位论文，1999．
[20]．叶中行，林建忠．数理金融—资产定价与金融决策理论．科学出版社，北京，1998．
[21]．Hull．J．张陶伟译．期权、期货和其它衍生产品．华夏出版社，北京，2000．
[22]．候为波，徐成贤．证券组合M—V有效边缘动态分析，系统工程学报，2000，15(1)，26-31．
[23]．马永开，唐小我．限制性卖空条件下β值证券组合投资模型．系统工程学报，2001，16(2)，81—87．
[24]．谭浩强．C程序设计．清华大学出版社，北京，1995．
[25]．肖庆宪，郑祖康．股票价格过程方差函数的统计推断．应用概率统计2000，16(2)，182-190．
[26]．杨昭军，邹捷中．一种亚式期权的套期保值策略．应用数学学报，2001，24(1)，56-60．
[27]．薛红，聂赞坎．Multi-dimensional black-sholes model with non-constant coefficients and dividend．应用数学，2000，13(3)，133-138．
[28]．刘忠．SV模型下的期权定价和风险计量．应用概率统计，2000，16(4)，365-370．
[29]．候为波，徐成贤．证券组合投资决策的β模型．应用数学，2000，13(3)，25-30．
[30]．曾勇，唐小我，郑维敏．基金分离定理的进一步研究．系统工程学报，2001，16(3)，187-191．
[31]．荣喜民，张世英．再保险定价的研究．系统工程学报，2001，16(6)，471-474．
[32]．何基报，茆诗松．对数正态分布场合无失效的Bayes验证试方案．应用概率统计，2000，16(3)，239-248．
[33]．林金官．心理状态数的Bayes估计．数理统计与管理，2001，20(2)，31-35．
[34]．俞迎达．Black-scholes期权定价模型的简化推导．数学的实践与认识，2001，31(6)，756-758．
[35]．Anold．V．I．Ordinary differential equations．沈家骐，周宝熙，卢亭鹤译．科学出版社，北京，2001．
[36]．盛昭瀚，马军海．非线性动力系统分析引论．科学出版社，北京，2001．
[37]．中国证券业协会．证券投资分析．上海财经大学出版社，上海，2002．
[38]．中国证券业协会．证券市场基础知识．上海财经大学出版社，上海，2002．
[39]．钱敏平，龚光鲁．随机过程论，北京大学出版社，北京，1997．
[40]．Sharpe．W．F．Portfolio theory and capiml markets．霍小虎，奚敬泽，王继德，郑启鸣译．中国经济出版社，北京，1992．