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@ARTICLE{RydinGorjo:891975,
      author       = {Rydin Gorjão, Leonardo and Witthaut, Dirk and Lehnertz,
                      Klaus and Lind, Pedro G.},
      title        = {{A}rbitrary-{O}rder {F}inite-{T}ime {C}orrections for the
                      {K}ramers–{M}oyal {O}perator},
      journal      = {Entropy},
      volume       = {23},
      number       = {5},
      issn         = {1099-4300},
      address      = {Basel},
      publisher    = {MDPI},
      reportid     = {FZJ-2021-01852},
      pages        = {517 -},
      year         = {2021},
      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.},
      cin          = {IEK-STE},
      ddc          = {510},
      cid          = {I:(DE-Juel1)IEK-STE-20101013},
      pnm          = {111 - Energiesystemtransformation (POF4-111)},
      pid          = {G:(DE-HGF)POF4-111},
      typ          = {PUB:(DE-HGF)16},
      pubmed       = {33923154},
      UT           = {WOS:000653859500001},
      doi          = {10.3390/e23050517},
      url          = {https://juser.fz-juelich.de/record/891975},
}