On the number of crossings and bouncings of a diffusion at a sticky threshold

Abstract

In this paper, we study the asymptotic behavior of the number of crossings by a one-dimensional diffusion of a threshold where the process exhibits stickiness. We distinguish three types of crossings and show that to each type corresponds a distinct asymptotic regime for the respective number of crossings statistic. We introduce notions of bouncing as the symmetric counterparts to crossings and show that the corresponding number of bouncings statistics share the same asymptotic properties as their crossings counterparts. We first prove the results for sticky Brownian motion, then extend them to sticky-reflected Brownian motion (where only bouncing is possible) and to sticky diffusions. As an application, we propose consistent estimators for the stickiness parameter of sticky diffusions and sticky-reflected Brownian motion.