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.
