Calibration of Lévy processes using optimal control of Kolmogorov equations with periodic boundary conditions

    Mario Annunziato Affiliation
    ; Hanno Gottschalk Affiliation


We present an optimal control approach to the problem of model calibration for Lévy processes based on an non-parametric estimation procedure of the measure. The optimization problem is related to the maximum likelihood theory of sieves [25] and is formulated with the Fokker-Planck-Kolmogorov approach [3, 4].

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) [12]. The first order necessary 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 using a scheme composed of Chang-Cooper, BDF2 and direct quadrature methods, jointly to a non-linear conjugate gradient method. For the numerical solver of the Kolmogorov's forward equation we prove conditions for non-negativity and stability in the L1 norm of the discrete solution.

Keyword : optimal control of PIDE, Kolmogorov-Fokker-Planck equation, Lévy processes, non-parametric maximum likelihood method, IMEX numerical method

How to Cite
Annunziato, M., & Gottschalk, H. (2018). Calibration of Lévy processes using optimal control of Kolmogorov equations with periodic boundary conditions. Mathematical Modelling and Analysis, 23(3), 390-413.
Published in Issue
Jun 14, 2018
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