Generalized Langevin equation driven by Lévy processes: A probabilistic, numerical and time series based approach

Abstract. Recently, various models have been developed, including the fractional Brownian motion (fBm), to analyse the stochastic properties of geodetic time series together with the estimated geophysical signals. The noise spectrum of these time series is generally modelled as a mixed spectrum, with a sum of white and coloured noise. Here, we are interested in modelling the residual time series after deterministically subtracting geophysical signals from the observations. This residual time series is then assumed to be a sum of three stochastic processes, including the family of Lévy processes. The introduction of a third stochastic term models the remaining residual signals and other correlated processes. Via simulations and real time series, we identify three classes of Lévy processes, namely Gaussian, fractional and stable. In the first case, residuals are predominantly constituted of short-memory processes. The fractional Lévy process can be an alternative model to the fBm in the presence of long-term correlations and self-similarity properties. The stable process is here restrained to the special case of infinite variance, which can be only satisfied in the case of heavy-tailed distributions in the application to geodetic time series. Therefore, the model implies potential anxiety in the functional model selection, where missing geophysical information can generate such residual time series.

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Application of Levy Processes in Modelling (Geodetic) Time Series With Mixed Spectra

10.5194/npg-2019-48 ◽

2019 ◽

Author(s):

Jean-Philippe Montillet ◽

Xiaoxing He ◽

Kegen Yu

Keyword(s):

Time Series ◽

Lévy Processes ◽

Stable Process ◽

Functional Model ◽

Noise Spectrum ◽

Levy Processes ◽

Residual Time ◽

Heavy Tailed Distributions ◽

First Case ◽

Heavy Tailed

Abstract. Recently, various models have been developed, including the fractional Brownian motion (fBm), to analyse the stochastic properties of geodetic time series, together with the extraction of geophysical signals. The noise spectrum of these time series is generally modeled as a mixed spectrum, with a sum of white and coloured noise. Here, we are interested in modelling the residual time series, after deterministically subtracting geophysical signals from the observations. This residual time series is then assumed to be a sum of three random variables (r.v.), with the last r.v. belonging to the family of Levy processes. This stochastic term models the remaining residual signals and other correlated processes. Via simulations and real time series, we identify three classes of Levy processes: Gaussian, fractional and stable. In the first case, residuals are predominantly constituted of short-memory processes. Fractional Levy process can be an alternative model to the fBm in the presence of long-term correlations and self-similarity property. Stable process is characterized by a large variance, which can be satisfied in the case of heavy-tailed distributions. The application to geodetic time series implies potential anxiety in the functional model selection where missing geophysical information can generate such residual time series.

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Reconstruction of Complex Dynamical Systems Using Stochastic Differential Equations

10.5194/egusphere-egu2020-4103 ◽

2020 ◽

Author(s):

Forough Hassanibesheli ◽

Niklas Boers ◽

Jürgen Kurths

Keyword(s):

Time Series ◽

Differential Equations ◽

Complex Systems ◽

Stochastic Differential Equations ◽

Langevin Equation ◽

Degrees Of Freedom ◽

Evolution Equations ◽

Ice Core ◽

Dynamical Behavior ◽

Generalized Langevin Equation

A complex system is a system composed of highly interconnected components in which the collective property of an underlying system cannot be described by dynamical behavior of the individual parts. Typically, complex systems are governed by nonlinear interactions and intricate fluctuations, thus to retrieve dynamics of a system, it is required to characterize and asses interactions between deterministic tendencies and random fluctuations.&#160;For systems with large numbers of degrees of freedom, interacting across various time scales, deriving time-evolution equations from data is computationally expensive. A possible way to circumvent this problem is to isolate a small number of relatively slow degrees of freedom that may suffice to characterize the underlying dynamics and solve the governing motion equation for the reduced-dimension system in the framework of stochastic differential equations(SDEs). &#160;For some specific example settings, we have studied the performance of three stochastic dimension-reduction methods (Langevin equation(LE), generalized Langevin Equation(GLE) and Empirical Model Reduction(EMR)) to model various synthetic and real-world time series. In this study corresponding numerical simulations of all models have been examined by probability distribution function(PDF) and Autocorrelation function(ACF) of the average simulated time series as statistical benchmarks for assessing the differnt models' performance.&#160;First we reconstruct the Ni&#241;o-3 monthly sea surface temperature (SST) indices averages across (5&#176;N&#8211;5&#176;S, 150&#176;&#8211;90&#176;W) from 1891 to 2015 using the three aforementioned stochastic models. We demonstrate that all these considered models can reproduce the same skewed and heavy-tailed distributions of Ni&#241;o-3 SST, comparing ACFs, GLE exhibits a tendency towards achieving a higher accuracy than LE and EMR. A particular challenge for deriving the underlying dynamics of complex systems from data is given by situations of abrupt transitions between alternative states. We show how the Kramers-Moyal approach to derive drift and diffusion terms for LEs can help in such situations. A prominent example of such 'Tipping Events' is given by the Dansgaard-Oeschger events during previous glacial intervals. We attempt to obtain the statistical properties of high-resolution, 20yr average, &#948;18O and Ca+2&#160;collected from the same ice core from the NGRIP on the GICC05 time scale. Through extensive analyses of various systems, our results signify that stochastic differential equation models considering memory effects are comparatively better approaches for understanding &#160;complex systems.&#160;

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Lévy copulae for financial returns

Dependence Modeling ◽

10.1515/demo-2016-0017 ◽

2016 ◽

Vol 4 (1) ◽

Author(s):

Ostap Okhrin

Keyword(s):

Time Series ◽

Stock Returns ◽

Lévy Processes ◽

Levy Processes ◽

Time Series Models ◽

Financial Returns ◽

Levy Copula

AbstractThe paper uses Lévy processes and bivariate Lévy copulae in order to model the behavior of intraday log-returns. Based on assumptions about the form of marginal tail integrals and a Clayton Lévy copula, the model allows for capturing intraday cross-dependency. The model is applied to VaR of the portfolios constructed on stock returns as well as on cryptocurrencies. The proposed method shows fair performance compared to classical time series models.

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Modelling financial time series with Lévy processes

Financial Modelling with Jump Processes ◽

10.1201/9780203485217-13 ◽

2003 ◽

pp. 225-261

Keyword(s):

Time Series ◽

Lévy Processes ◽

Financial Time Series ◽

Levy Processes ◽

Financial Time

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Application of Lévy Processes in Modelling (Geodetic) Time Series With Mixed Spectra

10.5194/npg-2020-23 ◽

2020 ◽

Author(s):

Jean-Philippe Montillet ◽

Xiaoxing He ◽

Kegen Yu ◽

Changliang Xiong

Keyword(s):

Time Series ◽

Lévy Processes ◽

Stable Process ◽

Functional Model ◽

Noise Spectrum ◽

Levy Processes ◽

Infinite Variance ◽

Residual Time ◽

First Case ◽

Heavy Tailed

Abstract. Recently, various models have been developed, including the fractional Brownian motion (fBm), to analyse the stochastic properties of geodetic time series, together with the estimated geophysical signals. The noise spectrum of these time series is generally modelled as a mixed spectrum, with a sum of white and coloured noise. Here, we are interested in modelling the residual time series, after deterministically subtracting geophysical signals from the observations. This residual time series is then assumed to be a sum of three stochastic processes, including the family of Lévy processes. The introduction of a third stochastic term models the remaining residual signals and other correlated processes. Via simulations and real time series,we identify three classes of Lévy processes: Gaussian, fractional and stable. In the first case, residuals are predominantly constituted of short-memory processes. Fractional Lévy process can be an alternative model to the fBm in the presence of long-term correlations and self-similarity property. Stable process is here restrained to the special case of infinite variance, which can be only satisfied in the case of heavy-tailed distributions in the application to geodetic time series. Therefore, it implies potential anxiety in the functional model selection where missing geophysical information can generate such residual time series.

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COMPLEXITY AND GENERALIZED EXPONENTIAL RELAXATION: MEMORY VERSUS RENEWAL

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127408021920 ◽

2008 ◽

Vol 18 (09) ◽

pp. 2709-2716 ◽

Cited By ~ 3

Author(s):

PAOLO GRIGOLINI

Keyword(s):

Time Series ◽

Complex Systems ◽

Langevin Equation ◽

Survival Probability ◽

Mathematical Expression ◽

Real Data ◽

Generalized Langevin Equation ◽

Dissipation Process ◽

Fluctuation Dissipation ◽

Exponential Relaxation

We illustrate two distinct approaches to the Mittag–Leffler relaxation, as a mathematical expression suitable for the interpretation of real data produced by complex systems, and especially those of physiological interest. The first approach is based on interpreting the fluctuation–dissipation process under study as obtained via Subordination to the Ordinary Fluctuation–Dissipation (SOFD) process. The second approach rests on the Generalized Langevin Equation (GLE). We prove that in the real cases of truncated time series the two theories generate a survival probability in the form of a stretched exponential, and that this property makes it hard to assess if a given time series obeys the GLE or the SOFD prescription. Some conjectures are made on the possibility of distinguishing the GLE from the SOFD predictions through the analysis of a single time series.

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