Calibration of L\'evy Processes using Optimal Control of Kolmogorov Equations with Periodic Boundary Conditions

TL;DR

This paper introduces an optimal control method for calibrating Le9vy processes by solving a high-dimensional inverse problem using a spline-based nonparametric approach and Kolmogorov equations, with proven stability and boundary handling.

Contribution

It develops a novel optimal control framework for Le9vy process calibration using PDE discretization, spline approximation, and stability analysis, advancing nonparametric estimation techniques.

Findings

Effective spline discretization with AIC for basis selection

Stable numerical solution of Kolmogorov equations in $L^1$ norm

Boundary projection to a torus minimizes information loss

Abstract

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.