Sticky-threshold diffusions, local time approximation and parameter estimation

Sara Mazzonetto (1, 2) , Alexis Anagnostakis (3, 4)

1 PASTA - Processus aléatoires spatio-temporels et leurs applications
2 IECL - Institut Élie Cartan de Lorraine
3 AIRSEA - Mathematics and computing applied to oceanic and atmospheric flows
4 IPS - Inférence Processus Stochastiques

We study a class of high-frequency path functionals of diffusions with a singular threshold, including the case of sticky-reflection. The functionals are built upon a test function and a normalizing sequence. We prove convergence to the local time, advancing existing results on sticky, oscillating (regime-switching), and skew or reflecting diffusions. Notably, we consider any normalizing sequence that diverges slower than the observation frequency. Additionally, we allow for jointly sticky-oscillating-skew thresholds. Combining these findings with an approximation of the occupation time, we propose consistent estimators for skewness and stickiness parameters.

Keywords: stickiness; skewness; reflection; Local time approximation; occupation time statistics; Sticky Brownian motion; Skew Brownian Motion; Oscillating Brownian Motion; Threshold diffusion; oscillating diffusion coefficient

Domains: Probability [math.PR]; Statistics [math.ST]

Sara Mazzonetto, Alexis Anagnostakis. Sticky-threshold diffusions, local time approximation and parameter estimation. 2024. ⟨hal-04503682v2⟩