Statistical modeling of diffusion processes with free knot splines

We consider the nonparametric estimation of the drift coefficient in a diffusion type process in which the diffusion coefficient is known and the drift coefficient depends in an unknown manner on a vector of time-dependent covariates. Based on many continuous realizations of the process, the estimator is constructed using the method of maximum likelihood, where the maximization is taken over a finite dimensional estimation space whose dimension grows with the sample size n. We focus on estimation spaces of polynomial splines. We obtain rates of convergence of the spline estimates when the knot positions are prespecified but the number of knots increases with the sample size. We also give the rates of convergence for free knot spline estimates, in which the knot positions of splines are treated as free parameters that are determined by data.