Nonparametric estimation of the transition density in dependence of time

We construct nonparametric, least-square projection estimators of the transition density of a diffusion process in dependence of the time $t$ and a space variable $x$. Consider $N$ continuously or discretely observed paths solving a $d$-dimensional stochastic differential equation and calculate the estimator by minimizing a contrast over a product of finite dimensional spaces. Under mild assumptions ensuring the transition density exists, we derive risk bounds for this estimator. We utilize these bounds to calculate the rate of convergence depending on the regularity of the density. We also propose a model selection procedure to derive estimators of the optimal dimensions of one or both orthonormal families generating the finite dimensional spaces of approximation. Finally, we illustrate the performance of our estimator for Ornstein-Uhlenbeck and Cox-Ingersoll-Ross processes in a simulation study.