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

Jannesar, M.; Sadeghi, Ali; Meyer, Ernst; Jafari, G. R.

doi:10.1038/s41598-019-48345-4

Cited by 5 publications

(3 citation statements)

References 40 publications

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“…One of the common approaches to computing the Markov length is the Chapman-Kolmogorov equation (CKE) 5,6 . This approach has been applied to diverse fields such as turbulence 7,8 , rough surface 1,9,10 , financial markets [11][12][13][14] , biology 15 , signal processing 16 , and earthquakes 17 .…”

Section: Introductionmentioning

confidence: 99%

Generalization of Chapman-Kolmogorov equation to two coupled processes

Motahari,

Noudehi,

Jamali

et al. 2024

Preprint

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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.

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Section: Introductionmentioning

confidence: 99%

Generalization of Chapman-Kolmogorov equation to two coupled processes

Motahari,

Noudehi,

Jamali

et al. 2024

Preprint

Self Cite

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“…Subsequently, we will study the change of individual stock prices over time using the Brownian equation (also known as the Langevin equation), with a particular emphasis on examining the dependence of stock prices on the parameters of the Brownian equation. When dealing with a large number of stock prices, we will employ a multitude of equations to evolve the changes in each stock price and calculate variables such as the mean and variance [4,5].…”

Section: Introductionmentioning

confidence: 99%

The Stochastic Model of Stock Prices

2023

accaf

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The purpose of this research is to investigate the utilization of the Brownian model in the analysis of financial data. In the field of finance, the analysis of financial data is crucial for making investment decisions and managing risks. The classical stochastic process model, known as the Brownian model, has been extensively employed in this domain. This paper provides an overview of the Brownian model, discusses its advantages and limitations in analyzing financial data, and presents empirical research findings. These findings serve as valuable references for future studies. The primary focus of the Brownian model is to study the random fluctuations of a variable while adhering to a specific statistical pattern. In this study, we derive the Brownian equation and explore the meanings of the damping term and the random variable term within the equation. When the random variable exhibits complete randomness at different time points, it is referred to as white noise. However, when considering certain time correlation lengths, the solution to the Brownian equation becomes more intricate. We meticulously examine how different noise terms and damping terms influence the solutions of the equation. Furthermore, we establish connections between these variables and various financial variables, particularly stock prices, to gain a practical understanding of the Brownian equation. The evolution of stock prices under different parameters is graphically illustrated and analyzed in detail.

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“…The Kramers–Moyal equation serves as a stepping stone to adequately describe time-series data with both diffusive and discontinuous characteristics, but it is nevertheless challenged by finite-time sampling in real-world data. Recent applications of the Kramers–Moyal equation include brain [ 3 , 4 ] and heart dynamics [ 5 ], stochastic harmonic oscillators [ 6 ], renewable-energy generation [ 7 ], solar irradiance [ 8 ], turbulence [ 9 ], nano-scale friction [ 10 ], and X-ray imaging [ 11 , 12 ].…”

Section: Introductionmentioning

confidence: 99%

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

Gorjão

Witthaut

Lehnertz

et al. 2021

Entropy

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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.

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

Cited by 5 publications

References 40 publications

Generalization of Chapman-Kolmogorov equation to two coupled processes

Generalization of Chapman-Kolmogorov equation to two coupled processes

The Stochastic Model of Stock Prices

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

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