  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.
