  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.
