Maximum Likelihood Estimation for Stochastic Differential Equations with Random Effects

We consider $N$ independent stochastic processes $(X_i(t), t\in [0,T_i])$, $i=1,\ldots, N$, defined by a stochastic differential equation with drift term depending on a random variable $\phi_i$. The distribution of the random effect $\phi_i$ depends on unknown parameters which are to be estimated from the continuous observation of the processes $X_i$. We give the expression of the exact likelihood. When the drift term depends linearly on the random effect $\phi_i$ and $\phi_i$ has Gaussian distribution, an explicit formula for the likelihood is obtained. We prove that the maximum likelihood estimator is consistent and asymptotically Gaussian, when $T_i=T$ for all $i$ and $N$ tends to infinity. We discuss the case of discrete observations. Estimators are computed on simulated data for several models and show good performances even when the length time interval of observations is not very large.