最小Hellinger距离方法在扩散过程参数估计中的应用

发布时间：2018-01-11 03:22

本文关键词：最小Hellinger距离方法在扩散过程参数估计中的应用　出处：《南京理工大学》2017年硕士论文　论文类型：学位论文

【摘要】：扩散过程在数理金融领域中扮演着重要的角色,例如在利率期限结构理论和投资组合的选取以及资产定价、衍生物定价等这些领域中都要用到扩散过程。扩散过程在这些领域中的应用证明了扩散过程是描述金融市场的最具有吸引力的工具之一。随着当代数理金融学的发展以及扩散过程在现实金融市场的应用逐渐广泛,扩散过程的统计分析问题已经成为当下金融数学研究上的一个热点问题。但是,在大多数的参数估计当中,都是基于扩散过程的不变分布密度进行的。由于扩散过程不变分布不能很好地体现有限时间内过程的动态特征,且在很多情况下,大部分扩散过程不具有不变分布。而扩散过程的转移密度对于模型检验等问题更为重要。由于扩散过程具有马尔可夫性质,多数具有不变的转移密度,其转移密度能获得连续时间过程的全部动态特征,因此基于转移密度的估计比基于不变密度的估计更有意义。所以,为了研究扩散过程的参数估计问题,本文基于最小Hellinger距离的定义给出了利用转移密度构造的参数估计量。首先给出扩散过程不变分布密度和联合密度的非参数估计量以及估计量的性质(一致收敛性以及渐近正态性)。基于此,给出了转移密度的非参数估计量并研究了它的性质(一致收敛性以及渐近正态性)。然后通过最小化扩散过程的转移密度和该密度的一个非参数估计量之间的Hellinger距离构造扩散过程的参数统计量,实现这一统计量的一致收敛性和渐近正态性。最后为了强调说明该方法的可行性,本文将这一方法应用于几何Brown运动和CEV模型这两个例子给出模拟分析。
[Abstract]:Diffusion process plays an important role in the field of mathematical finance, such as the theory of interest rate term structure, the selection of portfolio and asset pricing. Diffusion processes are used in these fields, such as derivative pricing. The application of diffusion processes in these fields proves that diffusion processes are one of the most attractive tools for describing financial markets. The application of the development and diffusion process in the real financial market is becoming more and more extensive. The statistical analysis of diffusion process has become a hot issue in the current financial mathematics research. However, in most of the parameter estimation. It is based on the invariant distribution density of the diffusion process, because the invariant distribution of the diffusion process can not well reflect the dynamic characteristics of the process in finite time, and in many cases. Most diffusion processes do not have invariant distribution, but the transfer density of diffusion process is more important for model checking. Because diffusion process has Markov property, most of them have invariant transfer density. The transfer density can obtain all the dynamic characteristics of the continuous time process, so the estimation based on the transfer density is more meaningful than the estimation based on the invariant density. Therefore, in order to study the parameter estimation of diffusion process. In this paper, based on the definition of minimum Hellinger distance, a parameter estimator constructed by transfer density is given. Firstly, the nonparametric estimators of invariant distribution density and joint density of diffusion process and the estimator of the estimator are given. Nature (. Uniform convergence and asymptotic normality. The nonparametric estimator of transfer density is given and its properties (uniform convergence and asymptotic normality) are studied. The parameter statistics of the diffusion process are then constructed by minimizing the Hellinger distance between the transfer density of the diffusion process and a nonparametric estimate of the density. The uniform convergence and asymptotic normality of this statistic are realized. Finally, the feasibility of the method is emphasized. In this paper, we apply this method to geometric Brown motion and CEV model.
【学位授予单位】：南京理工大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：F224;F830.59

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