Paper detail

Calibration of Lévy Processes using Optimal Control of Kolmogorov Equations with Periodic Boundary Conditions

We present an optimal control approach to the problem of model calibration for Lévy processes based on a non parametric estimation procedure. The calibration problem is of considerable interest in mathematical finance and beyond. Calibration of Lévy processes is particularly challenging as the jump distribution is given by an arbitrary Lévy 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 Lévy process is modelled as independent observations of the stochastic process at some terminal time $T$. We use a generic spline discretization of the Lévy jump measure and select an adequate size of the spline basis using the Akaike Information Criterion (AIC). The numerical solution of the Lévy 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 Lévy process on the real line can be projected to a torus, where it again is a Lévy process. If the torus is sufficiently large, the loss of information is negligible.