Estimation for stochastic differential equations with mixed effects

Abstract

We consider the long-term behaviour of a one-dimensional mixed effects diffusion process with a multivariate random effect φ in the drift coefficient. We first study the estimation of the random variable φ based on the observation of one sample path on the time interval as T tends to infinity. The process is not Markov and we characterize its invariant distributions. We build moments and maximum likelihood-type estimators of the random variable φ which are consistent and asymptotically mixed normal with rate . Moreover, we obtain non-asymptotic bounds for the moments of these estimators. Examples with a bivariate random effect are detailed. Afterwards, the estimation of parameters in the distribution of the random effect from N i.i.d. processes is investigated. Estimators are built and studied as both N and tend to infinity. We prove that the convergence rate of estimators differs when deterministic components are present in the random effects. For true random effects, the rate of convergence is whereas for deterministic components, the rate is . Illustrative examples are given.