Calibration for Multivariate Lévy-Driven Ornstein-Uhlenbeck Processes with Applications to Weak Subordination

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 a likelihood function allowing for parameter estimation of such a process using Fourier inversion assuming that the innovation term is absolutely continuous. We further give a method for simulating the observations based on an approximation of the innovation term and prove its convergence. 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 with a mixed-type distribution, 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 our likelihood method can be applied to accurately estimate the parameters in both cases.