We propose a functional estimation procedure for homogeneous stochastic diﬀerential equations based on a discrete sample of observations and with minimal requirements on the data generating process. We show how to identify the drift and diﬀusion function in situations where one or the other function is considered a nuisance parameter. The asymptotic behavior of the estimators is examined as the observation frequency increases and as the time span lengthens (that is, we implement both inﬁll and long span asymptotics). We prove consistency and convergence to mixtures of normal laws, where the mixing variates depend on the chronological local time of the underlying process, that is the time spent by the process in the vicinity of a spatial point. The estimation method and asymptotic results apply to both stationary and nonstationary processes.
Bandi, Federico M. and Phillips, Peter C.B., "Fully Nonparametric Estimation of Scalar Diffusion Models" (2001). Cowles Foundation Discussion Papers. 1594.