Functional convergence to the local time of a sticky diffusion

We establish the consistency of a local time approximation of a diffusion at a sticky threshold based on high-frequency observations. First, we prove the result for sticky Brownian motion, and then extend it to It\^o diffusions with a sticky point (SID). For this, we derive the pathwise formulation of an SID along with respective versions of key stochastic calculus results (It\^o formula, Girsanov theorem). Based on the local time approximation, we develop a consistent estimator for the stickiness parameter. We conclude with numerical experiments and assess statistical properties of the stickiness estimator and the local time approximation.