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Stochastic processes are encountered in many contexts, ranging fromgeneration sizes of bacterial colonies and service times in a queueing systemto displacements of Brownian particles and frequency fluctuations in anelectrical power grid. If such processes are Markov, then their probabilitydistribution is governed by the Kramers-Moyal (KM) equation, a partialdifferential equation that involves an infinite number of coefficients, whichdepend on the state variable. The KM coefficients must be evaluated based onmeasured time series for a data-driven reconstruction of the governingequations for the stochastic dynamics. We present an accurate method ofcomputing the KM coefficients, which relies on computing the coefficients'conditional moments based on the statistical moments of the time series. Themethod's advantages over state-of-the-art approaches are demonstrated byinvestigating prototypical stochastic processes with well-known properties.

Data-Driven Reconstruction of Stochastic Dynamical Equations based on Statistical Moments