  We address the problem of likelihood based inference for correlated diffusion
processes using Markov chain Monte Carlo (MCMC) techniques. Such a task
presents two interesting problems. First, the construction of the MCMC scheme
should ensure that the correlation coefficients are updated subject to the
positive definite constraints of the diffusion matrix. Second, a diffusion may
only be observed at a finite set of points and the marginal likelihood for the
parameters based on these observations is generally not available. We overcome
the first issue by using the Cholesky factorisation on the diffusion matrix. To
deal with the likelihood unavailability, we generalise the data augmentation
framework of Roberts and Stramer (2001 Biometrika 88(3):603-621) to
d-dimensional correlated diffusions including multivariate stochastic
volatility models. Our methodology is illustrated through simulation based
experiments and with daily EUR /USD, GBP/USD rates together with their implied
volatilities.
