Nonparametric estimation for stochastic differential equations with random effects

We consider N independent stochastic processes (Xj(t), t ∈ [0, T ]), j = 1, . . . , N , defined by a one-dimensional stochastic differential equation with coefficients depending on a random variable φj and study the nonparametric estimation of the density of the random effect φj 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-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, adaptive estimators are built. Estimators are implemented on simulated data for several examples. December 5, 2012