  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.
