Functional convergence to the local time of a sticky diffusion

We prove the convergence of a class of high frequency path-functionals of a sticky diffusion to its local time. First, we prove this for the sticky Brownian motion. Then, we extend the result to sticky stochastic differential equations. We combine the local time approximation with an approximation of the occupation time to set up a consistent stickiness estimator. Last, we perform numerical experiments to assess the properties of the stickiness estimator and the local time approximation.