We construct a generalization of the Ornstein-Uhlenbeck processes on the cone of covariance matrices endowed with the Log-Euclidean and the Affine-Invariant metrics. Our development exploits the Riemannian geometric structure of symmetric positive definite matrices viewed as a differential manifold. We then provide Bayesian inference for discretely observed diffusion processes of covariance matrices based on an MCMC algorithm built with the help of a novel diffusion bridge sampler accounting for the geometric structure. Our proposed algorithm is illustrated with a real data financial application.
翻译:我们将Ornstein-Uhlenbeck过程概括化为Ornstein-Uhlenbeck过程,用于配有Log-Euclidean和Affine-Invilant指标的共变矩阵锥体。我们的发展利用了Riemannian的对称正数确定矩阵的几何结构,这种结构被视为一个差异的多元。然后,我们用一种新的传播桥取样器对几何结构进行核算,根据一种MCMC算法,我们用真实的数据财务应用来说明我们提议的共变矩阵的分离观测扩散过程。