Local Nonparametric Estimation of Scalar Diffusions

This paper studies the functional estimation of the drift and diffusion functions for recurrent scalar diffusion processes from equally spaced observations using the local polynomial kernel approach. Almost sure convergence and a CLT for the estimators are established as the sampling frequency and the time span go to infinity. The asymptotic distributions follow a mixture of normal laws. This theory covers both positive and null recurrent diffusions. Almost sure convergence rates are sometimes path dependent but expected rates can always be characterized in terms of regularly varying functions. The general theory is specialized for positive recurrent diffusion processes, and it is shown in this case that the asymptotic distributions are normal. We also obtain the limit theory for kernel density estimators when the process is positive recurrent, namely, requiring only that the invariant probability measure exists. Nonetheless, it is also shown that such an estimator paradoxically vanishes almost surely when the invariant measure is fat tailed and nonintegrable, that is, in the null recurrent case.