Parameter Estimate Of Diffusion Processes Based On Discretely Observed Sample

Diffusion process is a powerful tool describing random phenomena. Its theory was founded on the1950s. Since then, it has received a lot of academic attention. Under the tireless efforts of many scholars, theories and methods have been considerable developed and successively used in many fields. However, statisticsof the process hasn’t attracted enough attention and theories and methods were in shortage in a long time, which greatly limits the application of theprocess.Due to the characteristics of the process, Statistical inference based on continuous observations is not difficult. However, the situation is very different in things based on discrete observations. This is mainly because in general model, the sample likelihood function has no analytic solution nor no simple and accurate approximation. Likelihood based statistical inference will face a lot of difficulty in this problem. Meanwhile, all samples people obtain are discretely observed. Research on this problem is important and urgent. As the core and basic of statistical inference, parameter estimate is undoubtedly one of the issues that people are most concerned about this and currently the focus of related research.This page first introduces the existing systems, the successful solution to the problem. Based on this, the page proposes two new ideas to the solution to this problem:Firstly, inserting a series of auxiliary time points into the time interval of the adjacent time points in the sampleto build an extended sample, on which Euler approximation will perform as a good approximation of the likelihood function of the original sample. Solving the maximum likelihood estimation problem on this approximation gives the estimates. It is a estimating problem based on incomplete sample. EM algorithm is a natural choice. Expectations in the algorithm can be estimated by simulation. Secondly, do not seek the approximation oflikelihood function, but to use approximated Bayes calculation method to convert it into a problem of acquiring a sample path given the parameters of the process. Then there are many efficient time-partition based numerical solution methods which can be employed. In order to improve the convergence speed, use calculations under more rough partition to guide the calculation. Results of different partitions are combined using Richardson extrapolation. Following these two ideas, the page establishes two new algorithms——Simulated-EM algorithm and approximated Bayesian estimation algorithm. Finally, numerical experiments shows the feasibility and effectiveness of the two algorithms.