On the Stratonovich Estimator for the Itô Diffusion

Abstract

For the parameter appearing non-linearly in the drift coefficient of homogeneous Itô stochastic differential equation having a stationary ergodic solution, the paper obtains the strong consistency of an approximate maximum likelihood estimator based on Stratonovich type approximation of the continuous Girsanov likelihood, under some regularity conditions, when the corresponding diffusion is observed at equally spaced dense time points over a long time interval in the high frequency regime. Pathwise convergence of stochastic integral approximations and their connection to discrete drift estimators is studied. Often it is shown that discrete drift estimators converge in probability. We obtain convergence of the estimator with probability one. Ornstein-Uhlenbeck process is considered as an example.