Sticky-threshold diffusions, local time approximation and parameter estimation

Abstract: We study a class of high frequency path functionals of a diffusion with a singular threshold, including the case of a sticky-reflection, and establish convergence to the local time. These functionals are built upon a test function and a normalizing sequence. This advances existing results on sticky, oscillating (regime-switching) and skew or reflecting diffusions in several directions. First, it considers any normalizing sequence that diverges slower than $n$. Second, it establishes the result for a sticky-oscillating-skew (SOS) threshold. Based on this, and an approximation of the occupation time, we devise consistent skew and stickiness parameter estimators.