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

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.

Change of Persistence in European Electricity Spot Prices

“…In principle, the Kramers-Moyal coefficients are defined in the limit τ → 0. In practical applications the resolution is limited, such that we resort to the smallest increments in the data set [72][73][74].…”

Definiteness of certain differential operators obtained via moment expansion of non‐negative functions

“…Like the approximation of nonlocal material laws, the Kramers-Moyal expansion with which the Fokker-Planck equation is obtained from a master equation [8] is a moment expansion. A large majority of examples of this expansion is truncated after the second term, which leads to the Fokker-Planck equation, while studies of higher-order truncations have not been overly frequent (see, e.g., [9] or the recent [10]), possibly because the difficulty of having to work with (and measure) a significantly higher number of parameters outweighs the benefits of additional details of the model in most situations, but also due to the Pawula theorem [11]. The asymptotic nature of the expansion means, however, that one need not fear Pawula's 'logical inconsistency' too much.…”