金融连续时间随机过程的统计推断

Continuous time stochastic processes defined by stochastic differential equations have long been used to model dynamic stochastic systems arising in Finance.They have been the focus of mathematical and probabilistic research for many decades. One latest surge of intereston these processes comes from financial econometrics and risk management. The most eminent use of the continuous time stochastic processes in the last two decades has been in finance following the works of Merton (1971) and Black and Scholes (1973) which established the foundation of option pricing theory for modern finance. Analysts and practitioners of these continuous-time stochastic models are increasingly aware that no matter how rich and powerful these models are for modeling a stochastic system, their applicability will be largely limited if model parameter cannot be estimated and the validity of the models cannot be confirmed with empirical data. Research on such issues of statistical inference is yet to be fully developed. A major obstacle which has hindered the development of the statistical inference is that the complexity of these processes prevents the use of standard statistical inference tools for instance the maximum likelihood approach. This is only further enhanced by a new model--observation structure where the model is continuous in time but the observations are collected only at discrete time points.This project considers parameter estimation and model testing for a wide range of continuous-time processes used in stochastic modeling. It includes processes which have discontinuous sample paths, are multivariate, only partially observed and are subject to long range dependence. Despite being wide ranging, a common opportunity offered by these processes is that the conditional characteristic functions are available for commonly used processes in stochastic modeling, that is usually not the case for the transitional distribution functions. Utilizing this opportunity for both parameter estimation and model testing is the focus of this project.

金融随机微分方程和金融时间序列分别是金融计量学和统计学科中的重要研究分支。它们也是金融工程建模、金融风险管理的重要工具。随机微分方程与时间序列的交叉带来了全新的，也是随机过程统计学中的最前沿的研究问题。具体地说，尽管建立在随机微分方程基础上的金融随机动态模型是连续时间的，但观察到的样本却是离散的。这种连续时间模型与离散时间数据之间的不一致性对金融连续时间随机模型的参数估计及模型检验是一个很大的挑战。同时由于连续时间模型比传统的离散时间模型复杂得多，这使得传统的统计估计方法，如极大似然及最小二乘方法不能直接应用。本课题将对现有的近似极大似然估计的统计学性质进行全面的评价；同时提出一套全新的近似极大似然估计的方法，这一方法适用于更一般的多维过程模型转移密度函数的近似。我们将建立一个在条件特征函数基础上的广泛应用于由列维过程驱动的随机微分方程，及部分可观测的过程的参数估计方法、模型检验的统计推

随机微分方程和时间序列分别是概率论和数理统计学科中的重要研究分支。随机微分方程广泛的用于动态系统、金融工程的建模中发展也趋成熟。与此同时，时间序列也是统计学研究中一个历史悠久的课题，其在实际工作中的应用范围非常宽广。对于这样两个发展相对成熟的研究方向，二者的交叉为我们带来了全新的，也是随机过程统计学中的最前沿的也最具挑战的研究问题。 随机过程统计推断对相关的统计学研究提出了全新的挑战。具体地说，尽管建立在随机微分方程基础上的随机动态模型是连续时间的，但观察到的样本是离散的时间序列，这些观测可能有非常密集的时间间隔, 即所谓的高频数据（high frequency data)。这种连续时间模型与离散时间数据之间的不一致性对连续时间随机模型的参数估计及模型检验是一个很大的挑战。同时由于连续时间模型比传统的离散时间模型复杂得多，这使得传统的统计估计方法，如极大似然及最小二乘方法不能直接应用。我们需要提出新的随机微分方程模型参数估计的统计方法、新的模型的统计检验方法。本课题将首先对现有的近似极大似然估计的统计学性质进行全面的评价； 同时将提出一套全新的近似极大似然估计的方法，这一方法适用于更一般的多维过程模型转移密度函数的近似。 我们将建立一个在条件特征函数基础上的广泛应用于由列维过程驱动的随机微分方程，及部分可观测得过程的参数估计方法、模型检验的统计推断框架。 这些就是我们本课题的主要研究意义。