Sticky-threshold diffusions, local time approximation and parameter estimation

Abstract

We study a class of high-frequency path functionals for one-dimensional diffusions with singular thresholds or boundaries, allowing for skewness, discontinuities in the diffusion coefficient, and stickiness or sticky reflection. These functionals, originally developed for local-time approximation in non-singular diffusions, are constructed from a test function and a diverging normalizing sequence. We establish convergence to local time, regardless of the nature of the singular threshold features, thereby extending several recent results on specific cases. Notably, our framework allows for any normalizing diverging sequence that is $o(n)$, where $n$ is the observation frequency, and thresholds at which several singular behaviors occur simultaneously. Combining the local-time approximation with occupation-time approximations, we construct consistent estimators of the stickiness and skewness parameters that remain valid in the presence of jump-discontinuities in the diffusion coefficient; this solves the estimation problem under the simultaneous presence of all these singular features.