关于扩散方程漂移系数估计方法的改进

【作者】 朱文佳；

【导师】 陈萍；

【作者基本信息】 南京理工大学 ， 金融学， 2010， 硕士

【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。更多还原

【Abstract】 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.更多还原