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

设[x(T)，0 t T]是扩散过程的一条实现路径，其系数具有未知的多维参数u。这位首席调查者和他的学生对估计u的真实值很感兴趣。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。当观测时间相对较长时，他们将在合理的假设下求出U对真实参数u的收敛速度。由于在实际应用中，x(T)只能在离散的时间点上记录，所以他们对与模型离散化相关的误差估计问题特别感兴趣。校长调查员和他的学生的研究在实际应用中具有浓厚的兴趣。扩散过程的参数估计在实际应用中有很多例子。例如，它可能与军用导弹的痕迹有关，或者与股票的价格有关。所有这些模型都包含一些未知参数。此外，对真实路径的任何观察都包含错误。假设观测误差只能由一个常数c有界，那么无论时间离散化方法如何增加计算步骤，估计的误差都不能显著地减小到超过某一数值。换句话说，当识别到观测误差时，原理调查员和他的学生将找出计算机获得足够准确结果所需的最少步骤。这项研究具有潜在的军事和金融应用价值。

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.