Estimation of population parameters in stochastic differential equations with random effects in the diffusion coefficient

¹ AgroParisTech, UMR518, 75005 Paris, France
maud.delattre@agroparistech.fr
² UMR CNRS 8145, Laboratoire MAP5, Université Paris Descartes, Sorbonne Paris Cité, France
valentine.genon-catalot@parisdescartes.fr
³ Laboratoire Jean Kuntzmann, UMR CNRS 5224, Université Grenoble-Alpes, France
adeline.leclercq-samson@imag.fr

Received: 2 March 2014
Revised: 6 December 2014

Abstract

We consider N independent stochastic processes (X_i(t), t ∈ [0, T]), i = 1,...,N, defined by a stochastic differential equation with diffusion coefficients depending linearly on a random variable φ_i. The distribution of the random effect φ_i depends on unknown population parameters which are to be estimated from a discrete observation of the processes (X_i). The likelihood generally does not have any closed form expression. Two estimation methods are proposed: one based on the Euler approximation of the likelihood and another based on estimations of the random effects. When the distribution of the random effects is Gamma, the asymptotic properties of the estimators are derived when both N and the number of observations per component X_i tend to infinity. The estimators are computed on simulated data for several models and show good performances.