Asymptotic Properties of Estimators in Stochastic Differential Equations with Additive Random Effects
Alkreemawi Walaa Khazal *
School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei, 430074, P.R. China and Department of Mathematics, College of Science, Basra University, Basra, Iraq.
Alsukaini Mohammed Sari
School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei, 430074, P.R. China and Department of Mathematics, College of Science, Basra University, Basra, Iraq.
Wang Xiang Jun
School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei, 430074, P.R. China.
*Author to whom correspondence should be addressed.
Abstract
A stochastic differential equation (SDE) defined N independent stochastic processes (Xi (t), t ∈ [0,Ti]),i = 1, ..., N, the drift term depends on the random variable ɸi . The distribution of the random effect ɸi depends on unknown parameters. When the drift term is defined linearly on the random effect ɸi (additive random effect) and ɸi has Gaussian Distribution, we propose an alternative route to prove asymptotic properties of Maximum Likelihood Estimator (MLE) by verifying the regularity conditions required through existing relevant theorems. We consider the Bayesian approach to learn the hyper parameters and proving asymptotic properties of the posterior distribution of the hyper parameters in the SDE’s model.
Keywords: Asymptotic normality, consistency, maximum likelihood estimator, mixed effects stochastic differential equations, posterior normality, posterior consistency.