Maximum likelihood Estimation for Stochastic Differential Equations with two Random Effects in the Diffusion Coefficient

Abstract

We study n independent stochastic processes(x_i (t),t_iЄ[o,t₁ ],i=1,......n) defined by a stochastic differential equation with diffusion coefficients depending nonlinearly on a random variables  and  (the random effects).The distributions of the random effects Ñ„_i,and,μ_i and  depends on unknown parameters which are to be estimated from the continuous observations of the processes x_i (t) . When the distributions of the random effects Ñ„ ,μ, are Gaussian and exponential respectively, we obtained an explicit formula for the likelihood function and the asymptotic properties (consistency and asymptotic normality) of the maximum likelihood estimator (MLE) are derived when  tend to infinity.