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.
翻译:考虑一个多变量 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。 我们得出一个可能性函数, 允许使用 Fourier 的参数来估计这一过程, 假设创新术语是绝对连续的。 我们进一步给出一种方法, 模拟基于创新术语近似值的观测结果, 并证明其趋同。 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α- 伽马 进程给出的, 这一过程是使用弱的从属关系生成的变异伽马 进程。 在前一种情况下, 我们给出了对背景驱动 L\' evy 进程的背景的清晰描述, 导致一种混合型分布的创新术语, 以及一个单独的概率函数。 在后一种情况下, 我们展示创新术语是绝对连续的。 模拟研究的结果表明, 我们可能采用的方法来准确估计两种情况下的参数。