On Maximum Likelihood Estimation in Randomly Stopped Diffusion-Type Processes
综述了扩散型连续观测过程的最大似然理论，特别关注观测在随机停止时间结束时的参数估计问题，推导了似然函数并讨论了估计量的一致性和渐近正态性。
A review is given of recent work on maximum likelihood theory for continuously observed processes of the diffusion type. A brief survey of basic definitions and results on continuous time stochastic processes and, particularly, processes of the diffusion type is presented too. Some new results concerning the case when the process is observed in a time interval from zero to a stopping time are included. Estimation of parameters determining the drift coefficient is considered. The relevant likelihood function and conditions for its existence are derived, and the score function as well as the observed and conditional Fisher information are discussed briefly. Particular interest is given to processes for which the drift coefficient depends linearly on the parameters such that the model is a curved exponential family. An expression for the affine ancillary is found and results on consistency and asymptotic normality of the maximum likelihood estimator are generalized to the case of an increasing sequence of stopping times tending to infinity or for which the observed information tends to infinity. Some convenient properties of the stopping rule defined by terminating the observation when a prescribed level of information is achieved are given, and finally, two examples are discussed.