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

Abstract: We distinguish between three types of crossing behaviors at a sticky threshold for a one-dimensional sticky diffusion process and analyze the asymptotic properties of the three corresponding number of crossings statistics in case the process is sticky Brownian motion. To each type of crossings corresponds a distinct asymptotic regime of the statistic. Additionally, we consider three types of bouncing behavior as symmetric counterparts to crossing behaviors. The number of bouncings behaves similarly to its number of crossings counterpart when the process is sticky Brownian motion. From this we deduce the behavior of the number of bouncings of sticky-reflected Brownian motion. The results on crossings and bouncings are extended to sticky diffusions. As an application, we propose consistent estimators for the stickiness parameter of sticky diffusions and sticky-reflected Brownian motion.