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结构,这种结构被视为一个差异的多元。然后,我们为根据在对几何结构进行新的扩散桥取样器的帮助下建立的MCMCMC算法而分别观测到的共变矩阵扩散过程提供了巴伊西亚推论。我们提议的算法用真实的数据财务应用来说明。