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

We consider N independent stochastic processes (Xi(t), t ∈ 0, T), i = 1,...,N, defined by a stochastic differential equation with diffusion coefficients depending linearly on a random variable φi. The distribution of the random effect φi depends on unknown population parameters which are to be estimated from a discrete observation of the processes (Xi). 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 Xi tend to infinity. The estimators are computed on simulated data for several models and show good performances.