Detecting Stochasticity in Discrete Signals via Nonparametric Excursion Theorem
arXiv:2601.06009v1 Announce Type: new Abstract: We develop a practical framework for distinguishing diffusive stochastic processes from deterministic signals using only a single discrete time series. Our approach is based on classical excursion and crossing theorems for continuous semimartingales, which correlates number $N_varepsilon$ of excursions of magnitude at least $varepsilon$ with the quadratic variation $[X]_T$ of the process. The scaling law holds universally for all continuous semimartingales with finite quadratic variation, including general Ito diffusions with nonlinear or state-dependent […]