A Langevin equation that governs the irregular stick-slip nano-scale friction

Chapman-Kolmogorov test for estimating memory length of two coupled processes

Real-world processes often display a prolonged memory, which extends beyond the single-step dependency characteristic of Markov processes. In addition, the current state of an empirical process is often not only influenced by its own past but also by the past states of other dependent processes. This study introduces the generalized version of the Chapman-Kolmogorov equation (CKE) to estimate the memory size in such scenarios. To assess the applicability of our approach, we generate coupled time series with predetermined memory lengths using the autoregressive model. The results show a high degree of accuracy in measuring memory lengths. Subsequently, we employ the generalized CKE to analyze cryptocurrency data as a real-world case study. Our results indicate that the past dynamics of cryptocurrencies significantly impact their current states, thereby highlighting interdependencies among them. The method proposed in this study can be also utilized in forecasting coupled time series.

Arbitrary-Order Finite-Time Corrections for the Kramers-Moyal Operator

With the aim of improving the reconstruction of stochastic evolution equations from empirical time-series data, we derive a full representation of the generator of the Kramers-Moyal operator via a power-series expansion of the exponential operator. This expansion is necessary for deriving the different terms in a stochastic differential equation. With the full representation of this operator, we are able to separate finite-time corrections of the power-series expansion of arbitrary order into terms with and without derivatives of the Kramers-Moyal coefficients. We arrive at a closed-form solution expressed through conditional moments, which can be extracted directly from time-series data with a finite sampling intervals. We provide all finite-time correction terms for parametric and non-parametric estimation of the Kramers-Moyal coefficients for discontinuous processes which can be easily implemented-employing Bell polynomials-in time-series analyses of stochastic processes. With exemplary cases of insufficiently sampled diffusion and jump-diffusion processes, we demonstrate the advantages of our arbitrary-order finite-time corrections and their impact in distinguishing diffusion and jump-diffusion processes strictly from time-series data.