Truncated dynamics and estimation of diffusion equations

We study inference on continuous-time processes from discrete data with a given time interval between consecutive observations, and propose a modification of the sieve estimation method based on the infinitesimal generator. Our approach consists on truncating the initial process to improve the estimation of the eigenfunctions at the boundaries of the set of admissible values. For diffusion processes, nonparametric estimation of the drift and volatility are derived. A prior truncation is also useful to eliminate in practice the specific dynamics of extreme risks.