Calibration for multivariate L\'evy-driven Ornstein-Uhlenbeck processes with applications to weak subordination

TL;DR

This paper develops a likelihood-based calibration method for multivariate Lévy-driven Ornstein-Uhlenbeck processes, focusing on cases involving weak variance alpha-gamma processes, enabling exact simulation and accurate parameter estimation.

Contribution

It derives explicit likelihood functions for these processes and demonstrates their practical application through simulation, advancing the calibration of complex Lévy-driven models.

Findings

Likelihood functions can be explicitly derived for these processes.

Maximum likelihood estimation via Fourier inversion is effective.

Exact simulation is feasible for processes with weak variance alpha-gamma background Lévy processes.

Abstract

Consider a multivariate L\'evy-driven Ornstein-Uhlenbeck process where the stationary distribution or background driving L\'evy process is from a parametric family. We derive the likelihood function assuming that the innovation term is absolutely continuous. Two examples are studied in detail: the process where the stationary distribution or background driving L\'evy process is given by a weak variance alpha-gamma process, which is a multivariate generalisation of the variance gamma process created using weak subordination. In the former case, we give an explicit representation of the background driving L\'evy process, leading to an innovation term which is discrete and continuous mixture, allowing for the exact simulation of the process, and a separate likelihood function. In the latter case, we show the innovation term is absolutely continuous. The results of a simulation study demonstrate that maximum likelihood numerically computed using Fourier inversion can be applied to accurately estimate the parameters in both cases.