Likelihood-based inference for a class of multivariate diffusions with unobserved paths

This paper presents a Markov chain Monte Carlo algorithm for a class of multivariate diffusion models with unobserved paths. This class is of high practical interest as it includes most diffusion driven stochastic volatility models. The algorithm is based on a data augmentation scheme where the paths are treated as missing data. However, unless these paths are transformed so that the dominating measure is independent of any parameters, the algorithm becomes reducible. The methodology developed in Roberts and Stramer [2001a. On inference for partial observed nonlinear diffusion models using the metropolis-hastings algorithm. Biometrika 88(3); 603–621] circumvents the problem for scalar diffusions. We extend this framework to the class of models of this paper by introducing an appropriate reparametrisation of the likelihood that can be used to construct an irreducible data augmentation scheme. Practical implementation issues are considered and the methodology is applied to simulated data from the Heston model.