  Time homogeneous polynomial processes are Markov processes whose moments can
be calculated easily through matrix exponentials. In this work, we develop a
notion of time inhomogeneous polynomial processes where the coeffiecients of
the process may depend on time. A full characterization of this model class is
given by means of their semimartingale characteristics. We show that in
general, the computation of moments by matrix exponentials is no longer
possible. As an alternative we explore a connection to Magnus series for fast
numerical approximations.
  Time-inhomogeneity is important in a number of applications: in
term-structure models, this allows a perfect calibration to available prices.
In electricity markets, seasonality comes naturally into play and have to be
captured by the used models. The model class studied in this work extends
existing models, for example Sato processes and time-inhomogeneous affine
processes.
