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

With the aim of improving reconstruction stochastic evolution equations from empirical time-series data, we derive a full representation generator Kramers-Moyal operator via power-series expansion exponential operator. This is necessary for deriving different terms in differential equation. this operator, are able to separate finite-time corrections arbitrary order into with and without derivatives coefficients. We arrive at closed-form solution expressed through conditional moments, which can be extracted directly data finite sampling intervals. provide all correction parametric non-parametric estimation coefficients discontinuous processes easily implemented - employing Bell polynomials analyses processes. exemplary cases insufficiently sampled diffusion jump-diffusion processes, demonstrate advantages our arbitrary-order their impact distinguishing strictly data.