Statistical modeling of diffusion processes with free knot splines

by Charles J. Stone and Jianhua Z. Huang

Aabstract

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.