Nonparametric estimation of the transition density function for diffusion processes

Abstract

We assume that we observe $N\in N^{\ast }$ independent copies of a diffusion process on a time-interval $[0,2T]$. For a given time $t\in (0,T]$, we estimate the transition density $p_t(x,.)$, namely the conditional density of $X_{t+s}$ given $X_s=x$, under conditions on the diffusion coefficients ensuring that this quantity exists. We use a least squares projection method on a product of finite dimensional spaces, prove risk bounds for the estimator and propose an anisotropic model selection method, relying on several reference norms. A simulation study illustrates the theoretical part for Ornstein–Uhlenbeck or square-root (Cox-Ingersoll-Ross) processes.