Nonparametric estimation for stochastic differential equations with random effects

We consider N independent stochastic processes ( X j ( t ) , t ∈ [ 0 , T ] ) , j = 1 , … , N , defined by a one-dimensional stochastic differential equation with coefficients depending on a random variable ϕ j and study the nonparametric estimation of the density of the random effect ϕ j in two kinds of mixed models. A multiplicative random effect and an additive random effect are successively considered. In each case, we build kernel and deconvolution estimators and study their L 2 -risk. Asymptotic properties are evaluated as N tends to infinity for fixed T or for T = T ( N ) tending to infinity with N . For T ( N ) = N 2 , adaptive estimators are built. Estimators are implemented on simulated data for several examples.