Asymptotic Properties of Estimators in Stochastic Differential Equations with Additive Random Effects

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.