  Literature is full of inference techniques developed to estimate the
parameters of stochastic dynamical systems driven by the well-known Brownian
noise. Such diffusion models are often inappropriate models to properly
describe the dynamics reflected in many real-world data which are dominated by
jump discontinuities of various sizes and frequencies. To account for the
presence of jumps, jump-diffusion models are introduced and some inference
techniques are developed. Jump-diffusion models are also inadequate models
since they fail to reflect the frequent occurrence as well as the continuous
spectrum of natural jumps. It is, therefore, crucial to depart from the
classical stochastic systems like diffusion and jump-diffusion models and
resort to stochastic systems where the regime of stochasticity is governed by
the stochastic fluctuations of L\'evy type. Reconstruction of L\'evy-driven
dynamical systems, however, has been a major challenge. The literature on the
reconstruction of L\'evy-driven systems is rather poor: there are few
reconstruction algorithms developed which suffer from one or several problems
such as being data-hungry, failing to provide a full reconstruction of noise
parameters, tackling only some specific systems, failing to cope with
multivariate data in practice, lacking proper validation mechanisms, and many
more. This letter introduces a maximum likelihood estimation procedure which
grants a full reconstruction of the system, requires less data, and its
implementation for multivariate data is quite straightforward. To the best of
our knowledge this contribution is the first to tackle all the mentioned
shortcomings. We apply our algorithm to simulated data as well as an ice-core
dataset spanning the last glaciation. In particular, we find new insights about
the dynamics of the climate in the curse of the last glaciation which was not
found in previous studies.
