Detecting Stochasticity in Discrete Signals via Nonparametric Excursion Theorem

TL;DR

This paper introduces a nonparametric, theoretically grounded method to distinguish stochastic diffusive processes from deterministic signals using a single discrete time series, based on excursion counts and quadratic variation.

Contribution

It develops a universal, data-driven diffusion test leveraging excursion theorems, enabling robust classification of stochastic versus deterministic dynamics.

Findings

The method accurately classifies stochastic and deterministic signals in various systems.

It demonstrates robustness across canonical stochastic, chaotic, and periodic systems.

The approach is nonparametric and relies solely on the small-scale structure of semimartingales.

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 volatility, but fails sharply for deterministic systems -- thereby providing a theoretically-certfied method of distinguishing between these dynamics, as opposed to the subjective entropy or recurrence based state of the art methods. We construct a robust data-driven diffusion test. The method compares the empirical excursion counts against the theoretical expectation. The resulting ratio $K(\varepsilon)=N_{\varepsilon}^{\mathrm{emp}}/N_{\varepsilon}^{\mathrm{theory}}$ is then summarized by a log-log slope deviation measuring the $\varepsilon^{-2}$ law that provides a classification into diffusion-like or not. We demonstrate the method on canonical stochastic systems, some periodic and chaotic maps and systems with additive white noise, as well as the stochastic Duffing system. The approach is nonparametric, model-free, and relies only on the universal small-scale structure of continuous semimartingales.