  We present an optimal control approach to the problem of model calibration
for L\'evy processes based on a non parametric estimation procedure. The
calibration problem is of considerable interest in mathematical finance and
beyond. Calibration of L\'evy processes is particularly challenging as the jump
distribution is given by an arbitrary L\'evy measure, which form a infinite
dimensional space. In this work, we follow an approach which is related to the
maximum likelihood theory of sieves.
  The sampling of the L\'evy process is modelled as independent observations of
the stochastic process at some terminal time $T$. We use a generic spline
discretization of the L\'evy jump measure and select an adequate size of the
spline basis using the Akaike Information Criterion (AIC).
  The numerical solution of the L\'evy calibration problem requires efficient
optimization of the log likelihood functional in high dimensional parameter
spaces. We provide this by the optimal control of Kolmogorov's forward equation
for the probability density function (Fokker-Planck equation). The first order
optimality conditions are derived based on the Lagrange multiplier technique in
a functional space. The resulting Partial Integral-Differential Equations
(PIDE) are discretized, numerically solved and controlled using scheme a
composed of Chang-Cooper, BDF2 and direct quadrature methods. For the numerical
solver of the Kolmogorov's forward equation we prove conditions for
non-negativity and stability in the $L^1$ norm of the discrete solution. To set
boundary conditions, we argue that any L\'evy process on the real line can be
projected to a torus, where it again is a L\'evy process. If the torus is
sufficiently large, the loss of information is negligible.
