Bayesian model selection for partially observed diffusion models

We present an approach to Bayesian model selection for finitely observed diffusion processes. We use data augmentation by treating the paths between observed points as missing data. For a fixed model formulation, the strong dependence between the missing paths and the volatility of the diffusion can be broken down by adopting the method of Roberts & Stramer (2001). We describe how this method may be extended to the case of model selection via reversible jump Markov chain Monte Carlo. In addition we extend the formulation of a diffusion model to capture a potential non-Markov state dependence in the drift. Issues of appropriate choices of priors and efficient transdimensional proposal distributions for the reversible jump algorithm are also addressed. The approach is illustrated using simulated data and an example from finance.