Calibrating Multivariate L\'evy Processes with Neural Networks

TL;DR

This paper introduces a neural network-based approach for calibrating multivariate Lévy processes by approximating their densities, overcoming challenges of traditional methods in high dimensions and with less smooth densities.

Contribution

It proposes a novel neural network framework for nonparametric Lévy density estimation, improving robustness and accuracy over traditional piecewise methods.

Findings

Neural networks effectively capture sharp transitions in Lévy densities.

The method outperforms piecewise linear and radial basis function approaches.

Applicable to various nonparametric function estimation problems.

Abstract

Calibrating a L\'evy process usually requires characterizing its jump distribution. Traditionally this problem can be solved with nonparametric estimation using the empirical characteristic functions (ECF), assuming certain regularity, and results to date are mostly in 1D. For multivariate L\'evy processes and less smooth L\'evy densities, the problem becomes challenging as ECFs decay slowly and have large uncertainty because of limited observations. We solve this problem by approximating the L\'evy density with a parametrized functional form; the characteristic function is then estimated using numerical integration. In our benchmarks, we used deep neural networks and found that they are robust and can capture sharp transitions in the L\'evy density. They perform favorably compared to piecewise linear functions and radial basis functions. The methods and techniques developed here apply to many other problems that involve nonparametric estimation of functions embedded in a system model.