Résumé en anglais

Biological processes measured repeatedly among a series of individuals are standardly analyzed by mixed models. They use common regression function and error model, both depending on individual random parameters. Regression functions defined by parametric Stochastic Differential Equations (SDEs) model adequately biological processes. This results in a mixed-effects model defined by an SDE. We focus on the parameter maximum likelihood estimation of this model. As the likelihood is not explicit, we propose the use of a stochastic version of the Expectation- Maximization algorithm combined with the Particle Markov Chain Monte Carlo method. We prove the convergence of the proposed algorithm towards the maximum likelihood estimator. We illustrate the performance of this estimation method on simulated datasets. We consider two examples: the first one is based on an Ornstein-Uhlenbeck process with two random parameters and an additive error model; the second one is based on a time-inhomogeneous SDE (Gompertz SDE) with a stochastic volatility error model and three random parameters. We highlight the superiority of our estimator over an estimator based on the EM algorithm coupled with standard MCMC algorithms.