Rate of Convergence of Diffusion Processes--Application to Statistical Inference for Processes

设[x（t），0 t T]为系数中含有未知多维参数u的扩散过程的实现路径.主要研究者和他的学生对估计u的真实的值感兴趣。他们发现，即使u出现在扩散系数中，他们也可以定义未知u的最大似然估计U。当观测时间相对足够长时，他们将在合理的假设下找到U向真实参数u的收敛速度。由于在真实的应用中，x（t）只能在离散的时间点上被记录，因此他们对与模型离散化有关的误差估计问题特别感兴趣。 主要研究者和他的学生的研究有其利益的真实的应用。扩散过程参数估计的应用实例很多。例如，它可能与军事导弹的踪迹或股票的价格有关。所有这些模型都包含一些未知参数。此外，对真实的路径的任何观测都包含误差。假设观测误差只能由一个常数c限定，那么无论如何增加时间离散化方法的计算步数，估计误差都不能显著减小到某个数值以上。换句话说，主要研究者和他的学生将找出计算机在识别观察错误时获得足够准确结果所需的最小步骤。这项研究具有潜在的军事和金融应用。

0203823Zheng Let [x(t), 0 t T] be a realized path of a diffusion process with an unknown multi-dimensional parameter u in its coefficients. The Principle Investigator and his students are interested in estimating the real value of u. They discovered that they can define a Maximum Likelihood Estimator U of the unknown u even if u appeared in the diffusion coefficient. They are going to find the rate of convergence of U to the true parameter u under reasonable assumptions when the observation time is relatively long enough. Since in the real applications, x(t) can be only recorded at discrete time spots, they are particularly interested in the problems of error estimates related to discretizations of their models. The Principle Investigator and his students' research has its interests in the real applications. There are many examples of parameter estimates of diffusion processes in the applications. For example, it may be related to the trace of a military missile, or the price of a stock. All those models contain some unknown parameters. Moreover, any observation of the real path contains errors. Assuming that the observation error can be only bounded by a constant c, then no matter how one increases the computation steps in the time discretization method, the error of the estimate can not be significantly reduced beyond some number. In other words, the Principle Investigator and his students will find out the minimum steps needed for a computer to get enough accurate result when an observation error is recognized. This study has its potential military and financial applications.