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.
翻译:考虑一个多变L\'evy- ornstein- Uhlenbeck 进程, 固定分布或背景驱动 L\' evy 进程来自一个参数组。 我们得出的可能性函数假定创新术语是绝对连续的。 详细研究了两个例子: 固定分布或背景驱动 L\\' evy 进程是由一个微弱的变异阿尔法- 伽玛进程提供的, 这是一种对使用弱从属关系产生的差异伽玛进程的多变概括。 在前一种情况下, 我们给出了背景驱动 L\' evy 进程的明确描述, 导致一个独立和连续的混合的创新术语, 允许对过程进行精确的模拟, 以及一个单独的概率函数。 在后一种情况下, 我们显示创新术语是绝对连续的。 模拟研究的结果显示, 使用 Fourier Inversion 进行数字计算的最大可能性可以用于准确估计两个案例的参数 。