Abstract
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.
Highlights
The reconstruction of stochastic evolution equations from time-series data in terms of the Langevin equation and the corresponding Fokker–Planck equation is often challenged by the inevitably finite temporal sampling of time-series data
The Fokker–Planck equation is restricted to continuous stochastic processes, i.e., diffusion, and cannot adequately describe discontinuous transitions in time-series data
We expressed the exponential operator as a power series and worked out each element of the series, combining it in a series representation via the partial Bell polynomials
Summary
The reconstruction of stochastic evolution equations from time-series data in terms of the Langevin equation and the corresponding Fokker–Planck equation is often challenged by the inevitably finite temporal sampling of time-series data. We derive a full expansion of the generator of the Kramers–Moyal operator in exponential form for onedimensional Markovian processes This is equivalent to van Kampen’s system-size expansion, which is taken over a finite time interval τ [23,24]. We focus on the solution of this partial differential equation by representing the Kramers–Moyal operator in an exponential form and equating the conditional moments with the KM coefficients after representing the exponential operator as a power series This representation of the exponential operator can be used in other problems with an equivalent formulation [26,27,28] or similar discontinuous stochastic processes with different jump distributions, e.g., the Gamma distribution [29,30]
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