Statistical Inference for Stochastic Differential Equations from Discrete Observations

Abstract

We consider hybrid estimation for stochastic differential equations (SDEs) from discrete observations and adaptive estimation of a stochastic partial differential equation (SPDE) based on sample data. In order to obtain the maximum likelihood (ML) type estimators of discretely observed ergodic SDEs, we need a suitable initial value for optimization of the quasi log likelihood (QLL) function. The Bayes type estimators derived from both the reduced data and the thinned data are used as an initial estimator for optimization of the QLL function, and the ML type estimators are obtained by using the initial Bayes estimators. The ML type estimators with the initial Bayes type estimators are called the hybrid type estimators. The asymptotic properties of both the initial Bayes type and the hybrid type estimators are shown and the asymptotic behaviors of both the initial Bayes type and the hybrid type estimators are given through simulations. Next, we treat parametric estimation for a parabolic linear second-order SPDE with a small dispersion parameter based on high frequency data which are observed in time and space. The adaptive estimators of unknown parameters of the SPDE are obtained, and their asymptotic properties are investigated. Furthermore, we give the asymptotic behaviors of the adaptive estimators by simulations.