Estimation of the Diffusion Coefficient from Crossings

In a previous work we proposed a kernel method for estimating the value of a state‐dependent diffusion coefficient σ(x) from discrete time observations. We propose an estimator of the value σ(x), based on the squared length of the transitions where a crossing past the level x takes place and we investigate its limit properties. Using the Poisson equation and other limit theorems for discrete martingales, we prove consistency and asymptotic mixed normality of this estimator.ou