  We introduce a mean-reverting SDE whose solution is naturally defined on the
space of correlation matrices. This SDE can be seen as an extension of the
well-known Wright-Fisher diffusion. We provide conditions that ensure weak and
strong uniqueness of the SDE, and describe its ergodic limit. We also shed
light on a useful connection with Wishart processes that makes understand how
we get the full SDE. Then, we focus on the simulation of this diffusion and
present discretization schemes that achieve a second-order weak convergence.
Last, we explain how these correlation processes could be used to model the
dependence between financial assets.
