Nonparametric estimation for stochastic differential equations with random effects

International audienceWe consider $N$ independent stochastic processes $(X_j(t), t\in [0,T])$, $ j=1, \ldots,N$, defined by a one-dimensional stochastic differential equation with coefficients depending on a random variable $\phi_j$ and study the nonparametric estimation of the density of the random effect $\phi_j$ 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