Uniform and Lp Convergences of Nonparametric Estimation for Diffusion Models

We obtain the uniform convergence rates of nonparametric kernel estimators of the local time, the drift and volatility functions as well as their derivatives, of discretely sampled diffusion processes. Moreover, modified kernel estimators of the drift and volatility functions are considered and their Lp convergence rates are obtained. Our asymptotic results apply to recurrent diffusions which include both stationary or nonstationary cases. Our sampling scheme is two-dimensional, with sampling interval shrinking to zero and time span increasing to infinity jointly.