关于扩散方程漂移系数估计方法的改进
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摘要
本文提出了一种新的基于扩散过程轨道构造漂移系数样本的方法—对数增量法,通过理论及模拟分析说明了在适当条件下,特别是对于大多数金融数据,基于对数增量法获得的漂移系数估计量的收敛速度及有限样本性质均比基于传统的“直接增量法”所得到的结果要好。
本文首先回顾了扩散方程估计问题在理论和实证分析中的一些成果,然后针对扩散方程漂移系数的的非参数估计方法做了如下工作：
第二章主要讨论了漂移系数的非参数估计方法理论上的比较。运用Girsanov定理将漂移系数估计转化为非参数回归。对于一维函数f(x)估计量的构造,本文采用了非参数的核估计方法,并给出了估计量一致收敛性,相合性和正态性的一般结论。接着基于直接增量法提出了对数增量法,通过对比两种方法中的随机误差的方差来验证对数增量法比直接增量法更有效。
第三章研究几个具体化的模型：线性模型、非线性模型、OU随机波动率模型,分别用对数增量法和直接增量法这两种方法,对其漂移系数进行了非参数估计,并比较了两种方法获得的漂移系数估计量的收敛速度及有限样本性质,说明了对数增量法确实优于直接增量法.
第四章在具体化模型拟和较好的情况下,考虑无风险利率过程,对汇率数据进行了实证分析。
This paper proposed a new way of structuring drift coefficient observations based on the drift coefficient of diffusion process orbit observations-logarithmic incremental method,Through theoretical and simulate analysis, under appropriate conditions, especially for the majority of financial data, the convergence rate and the finite sample properties of drift coefficient estimator based on logarithmic incremental method are more superior than the result based on traditional direct incremental method.
In this paper, we first reviewed the fruit of diffusion estimation in theory and applications., then we do the following work about the nonparametric estimation problem in the drift coefficient of diffusion process.
In chapter 2, we mainly discussed the ways of nonparametric estimation of diffusion efficient. First we translated the problem of the nonparametric estimation of drift coefficient into the nonparametric regression by Girsanov theorem. For the structure of 1-dimentional function,we applied the kernel estimation on it. Further more we discussed the conclusion about the uniform convergence,consistency and asymptotic normality of it.Then we proposed the logarithmic incremental method based on the direct incremental method. It proved that logarithmic incremental method was more superior than the direct incremental method trough the comparation of the variance of random errors.
In chapter 3, we considered several specific models:linear model, nonlinear models and OU-Stochastic volatility model. Respectively we estimated the drift coefficient by logarithmic incremental method and direct incremental method, compared the convergence rate and the finite sample properties of the two methods. It showed that the logarithmic incremental method was indeed superior to the direct incremental method.
In chapter 4, Under the condition that the simulate result of the specific model was excellent, we considered the risk-free interest rate process and conducted an empirical analysis of exchange rate data.
本文首先回顾了扩散方程估计问题在理论和实证分析中的一些成果,然后针对扩散方程漂移系数的的非参数估计方法做了如下工作：
第二章主要讨论了漂移系数的非参数估计方法理论上的比较。运用Girsanov定理将漂移系数估计转化为非参数回归。对于一维函数f(x)估计量的构造,本文采用了非参数的核估计方法,并给出了估计量一致收敛性,相合性和正态性的一般结论。接着基于直接增量法提出了对数增量法,通过对比两种方法中的随机误差的方差来验证对数增量法比直接增量法更有效。
第三章研究几个具体化的模型：线性模型、非线性模型、OU随机波动率模型,分别用对数增量法和直接增量法这两种方法,对其漂移系数进行了非参数估计,并比较了两种方法获得的漂移系数估计量的收敛速度及有限样本性质,说明了对数增量法确实优于直接增量法.
第四章在具体化模型拟和较好的情况下,考虑无风险利率过程,对汇率数据进行了实证分析。
This paper proposed a new way of structuring drift coefficient observations based on the drift coefficient of diffusion process orbit observations-logarithmic incremental method,Through theoretical and simulate analysis, under appropriate conditions, especially for the majority of financial data, the convergence rate and the finite sample properties of drift coefficient estimator based on logarithmic incremental method are more superior than the result based on traditional direct incremental method.
In this paper, we first reviewed the fruit of diffusion estimation in theory and applications., then we do the following work about the nonparametric estimation problem in the drift coefficient of diffusion process.
In chapter 2, we mainly discussed the ways of nonparametric estimation of diffusion efficient. First we translated the problem of the nonparametric estimation of drift coefficient into the nonparametric regression by Girsanov theorem. For the structure of 1-dimentional function,we applied the kernel estimation on it. Further more we discussed the conclusion about the uniform convergence,consistency and asymptotic normality of it.Then we proposed the logarithmic incremental method based on the direct incremental method. It proved that logarithmic incremental method was more superior than the direct incremental method trough the comparation of the variance of random errors.
In chapter 3, we considered several specific models:linear model, nonlinear models and OU-Stochastic volatility model. Respectively we estimated the drift coefficient by logarithmic incremental method and direct incremental method, compared the convergence rate and the finite sample properties of the two methods. It showed that the logarithmic incremental method was indeed superior to the direct incremental method.
In chapter 4, Under the condition that the simulate result of the specific model was excellent, we considered the risk-free interest rate process and conducted an empirical analysis of exchange rate data.
引文
[1]Ho, T. S. Y. and Lee, S. B. (1986). Term structure movements and pricing interest rate contingent claims. J. Finance 41:1011-1029.
[2]Black, F., Derman, E. and Toy, W. (1990). A one-factor model of interest rates and its application to treasury bond options. Finan. Analysts'J.46:33-39.
[3]Black, F. and Karasinski, P. (1991). Bond and option pricing when short rates are lognormal. Finan. Analysts'J.47:52-59.
[4]Wong, E. (1971). Stochastic Processes in Information and Dynamical Systems. McGraw-Hill, New York.
[5]Duffie, D. (1996). Dynamic Asset Pricing Theory.2nd edition. Princeton University Press, Princeton, New Jersey.
[5]Merton, R. C. (1992). Continuous-Time Finance. Blackwell, Cambridge, Massachusetts.
[7]Vasicek,0. (1977). An equilibrium characterization of the term structure. J. Finan. Econom.5:177-188.
[8]Cox, J. C., Ingersoll, J. E. and Ross, S. A. (1980). An analysis of variable rate loan contracts. J. Finance 35:389-403.
[9]Cox, J. C., Ingersoll, J. E. and Ross, S. A. (1985). A theory of the term structure of interest rates. Econometrica 53:385-467.
[10]Chan, K. C., Karolyi, A. G., Longstaff, F. A. and Sanders, A. B. (1992). An empirical comparison of alternative models of the short-term interest rate. J.
[11]Yongmiao Hong and Haitao Li,Nonparametric specification testing for Continuous-Time Models with Applications to term structure of Interest Rates, The review of financial studies,18
[12]Johsn-Shnno(1987)Consistent nonparametric regression, Ann. Statist. 5,1977,595-645.
[13]Nadaraya, E. A., On estimating regression. Theory Prob. Appl.,1964, 9:141-142.
[14]Hull-white (1987,1988) Statistical inference of diffusion processes, Lecture Notes in Mathematics,5
[15]Gasser, T. and Muller, H.-G., Kernel estimation of regression functions, In Smoothing Techniques for Curve Estimation, Lecture Notes in Mathematics,757,1979,23-68. Springer-Verlag, New York
[16]K. C. Chan et.al., An empirical comparison of alternative models of the short-term interest rate, Journal of Finance, Volume47, Issue 3, Papers and Proceedings of the Fifty-Second Annual. Meeting of American Finance Association,.1992,6:1209-1227.
[17]Basawa, I. V.& Prakasa Rao, B. L. S, Statistical inference for stochastic processes, Academic Press, London.1980.
[19]Didier Dacunha-Castelle and Danielle Florens-Zmirou (1986). Estimation of the coefficientsof a diffusion from discrete observations. Stochastics,19:263-284.
[20]Lo, A. W. (1988):Maximum likelihood estimation of Generalized Ito Processes with discretely-sampled data, Econometric Theory,4: 231-247.
[21]Duffie, D., and K. Singleton (1993):Simulated Moments Estimation of Markov Models of Asset Prices, Econometrica,61:929-952.
[22]Zhiyan Xu and Weian Zheng, Statistical inference of diffusion processes,第七届概率统计年会报告文章,2002
[23]Basawa, I. V.& Prakasa Rao, B. L. S, Statistical inference for stochastic processes, Academic Press, London.1980.
[24]Florens-Zmirou, D, On estimation the diffusion coefficient from discrete observations, J. App, Prob,1993,30:790-804.
[25]Florens-Danielle, Estimation of the diffusion coefficient from crossings. Statistical Inference for stochastic processes 1999,1:175-195.
[26]Hoffmann. M., Adaptive estimation in diffusion process, Stochastic processes and their applications,1999,79:135-163,
[27]陈萍,随机波动率模型的统计推断及其衍生证券的定价,南京理工大学博士论文,2004.
[28]Jacod, J., Non-parametric Kernel Estimation of the Coefficient of a Diffusion, J. Scan. Stat.,2000,27(1):83-98
[29]Dennis Kristensen, Estimation in Two Classes of Semiparametric Diffusion Models,2004.
[30]Ait-Sahalia, Y. (1996a) Nonparametric Pricing of Interest Rate Derivative Securities. Econometrica 64:527-560.
[31]B. L. S. Prakasa Rao, Nonparametric Functional Estimation, Academic Press, Inc, London,1983
[32]Kutoyants, Yu, A, Parameter estimation for stochastic process, HeldermannVerlag, Berlin,1984.
[33]Fan and Gijbcls,I., Local palynomial modeling and its applications. Chapman & Hall, London.1996.
[34]Fan, J., Zhang, C. M. and Zhang, J. (2001). Generalized likelihood ratio statistics and Wilks phenomenon. Ann. Statist.29:153-193.
[35]陈萍,杨孝平,资产方程的非参数估计,南京理工大学学报,Vol.28,No.2,2004.
[36]Chen Ping, Wang Jinde, Wavelet estimation in time dependent diffusion models,2007,38 (6):1-16
[37]Stone, C.J., Consistent nonparametric regression, Ann. Statist., 1977,5:595-645.
[38]Nadaraya, E. A., On estimating regression,Theory Prob, Appl.,1964, 9:141-142.
[39]Gasser, T. and Muller, H.-G., Kernel estimation of regression functions, In Smoothing Techniques for Curve Estimation, Lecture Notes in Mathematics,1979,757:23-68.Springer-Verlag, New York.
[40]Pritsker, M., Nonparametric Density Estimation and Tests of Continuous Time Interest Rate Models, Review of Financial Studies, 1998,11:449-487.
[41]万成高,鞅的极限理论,科学出版社,2002.3
[42]Bent Oksendal, Stochastic Differential Equations, Springer-Verlag, 2000
[43]Federico M Bandi; Peter C B Phillips, Fully nonparametric estimation of scalar diffusion models, Econometrica, Jan 2003; 71:1;
[44]http://research. stlouisfed. org/fred2/categories/15
[2]Black, F., Derman, E. and Toy, W. (1990). A one-factor model of interest rates and its application to treasury bond options. Finan. Analysts'J.46:33-39.
[3]Black, F. and Karasinski, P. (1991). Bond and option pricing when short rates are lognormal. Finan. Analysts'J.47:52-59.
[4]Wong, E. (1971). Stochastic Processes in Information and Dynamical Systems. McGraw-Hill, New York.
[5]Duffie, D. (1996). Dynamic Asset Pricing Theory.2nd edition. Princeton University Press, Princeton, New Jersey.
[5]Merton, R. C. (1992). Continuous-Time Finance. Blackwell, Cambridge, Massachusetts.
[7]Vasicek,0. (1977). An equilibrium characterization of the term structure. J. Finan. Econom.5:177-188.
[8]Cox, J. C., Ingersoll, J. E. and Ross, S. A. (1980). An analysis of variable rate loan contracts. J. Finance 35:389-403.
[9]Cox, J. C., Ingersoll, J. E. and Ross, S. A. (1985). A theory of the term structure of interest rates. Econometrica 53:385-467.
[10]Chan, K. C., Karolyi, A. G., Longstaff, F. A. and Sanders, A. B. (1992). An empirical comparison of alternative models of the short-term interest rate. J.
[11]Yongmiao Hong and Haitao Li,Nonparametric specification testing for Continuous-Time Models with Applications to term structure of Interest Rates, The review of financial studies,18
[12]Johsn-Shnno(1987)Consistent nonparametric regression, Ann. Statist. 5,1977,595-645.
[13]Nadaraya, E. A., On estimating regression. Theory Prob. Appl.,1964, 9:141-142.
[14]Hull-white (1987,1988) Statistical inference of diffusion processes, Lecture Notes in Mathematics,5
[15]Gasser, T. and Muller, H.-G., Kernel estimation of regression functions, In Smoothing Techniques for Curve Estimation, Lecture Notes in Mathematics,757,1979,23-68. Springer-Verlag, New York
[16]K. C. Chan et.al., An empirical comparison of alternative models of the short-term interest rate, Journal of Finance, Volume47, Issue 3, Papers and Proceedings of the Fifty-Second Annual. Meeting of American Finance Association,.1992,6:1209-1227.
[17]Basawa, I. V.& Prakasa Rao, B. L. S, Statistical inference for stochastic processes, Academic Press, London.1980.
[19]Didier Dacunha-Castelle and Danielle Florens-Zmirou (1986). Estimation of the coefficientsof a diffusion from discrete observations. Stochastics,19:263-284.
[20]Lo, A. W. (1988):Maximum likelihood estimation of Generalized Ito Processes with discretely-sampled data, Econometric Theory,4: 231-247.
[21]Duffie, D., and K. Singleton (1993):Simulated Moments Estimation of Markov Models of Asset Prices, Econometrica,61:929-952.
[22]Zhiyan Xu and Weian Zheng, Statistical inference of diffusion processes,第七届概率统计年会报告文章,2002
[23]Basawa, I. V.& Prakasa Rao, B. L. S, Statistical inference for stochastic processes, Academic Press, London.1980.
[24]Florens-Zmirou, D, On estimation the diffusion coefficient from discrete observations, J. App, Prob,1993,30:790-804.
[25]Florens-Danielle, Estimation of the diffusion coefficient from crossings. Statistical Inference for stochastic processes 1999,1:175-195.
[26]Hoffmann. M., Adaptive estimation in diffusion process, Stochastic processes and their applications,1999,79:135-163,
[27]陈萍,随机波动率模型的统计推断及其衍生证券的定价,南京理工大学博士论文,2004.
[28]Jacod, J., Non-parametric Kernel Estimation of the Coefficient of a Diffusion, J. Scan. Stat.,2000,27(1):83-98
[29]Dennis Kristensen, Estimation in Two Classes of Semiparametric Diffusion Models,2004.
[30]Ait-Sahalia, Y. (1996a) Nonparametric Pricing of Interest Rate Derivative Securities. Econometrica 64:527-560.
[31]B. L. S. Prakasa Rao, Nonparametric Functional Estimation, Academic Press, Inc, London,1983
[32]Kutoyants, Yu, A, Parameter estimation for stochastic process, HeldermannVerlag, Berlin,1984.
[33]Fan and Gijbcls,I., Local palynomial modeling and its applications. Chapman & Hall, London.1996.
[34]Fan, J., Zhang, C. M. and Zhang, J. (2001). Generalized likelihood ratio statistics and Wilks phenomenon. Ann. Statist.29:153-193.
[35]陈萍,杨孝平,资产方程的非参数估计,南京理工大学学报,Vol.28,No.2,2004.
[36]Chen Ping, Wang Jinde, Wavelet estimation in time dependent diffusion models,2007,38 (6):1-16
[37]Stone, C.J., Consistent nonparametric regression, Ann. Statist., 1977,5:595-645.
[38]Nadaraya, E. A., On estimating regression,Theory Prob, Appl.,1964, 9:141-142.
[39]Gasser, T. and Muller, H.-G., Kernel estimation of regression functions, In Smoothing Techniques for Curve Estimation, Lecture Notes in Mathematics,1979,757:23-68.Springer-Verlag, New York.
[40]Pritsker, M., Nonparametric Density Estimation and Tests of Continuous Time Interest Rate Models, Review of Financial Studies, 1998,11:449-487.
[41]万成高,鞅的极限理论,科学出版社,2002.3
[42]Bent Oksendal, Stochastic Differential Equations, Springer-Verlag, 2000
[43]Federico M Bandi; Peter C B Phillips, Fully nonparametric estimation of scalar diffusion models, Econometrica, Jan 2003; 71:1;
[44]http://research. stlouisfed. org/fred2/categories/15