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

• Main Authors: Mario Annunziato, Hanno Gottschalk
• Format: Article
• Language: English
• Published: Vilnius Gediminas Technical University 2018-06-01
• Series: Mathematical Modelling and Analysis
• Subjects: optimal control of PIDE; Kolmogorov-Fokker-Planck equation; Lévy processes; non-parametric maximum likelihood method; IMEX numerical method
• Online Access: https://journals.vgtu.lt/index.php/MMA/article/view/2814
• ISSN: 1392-6292; 1648-3510

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.