‘Purposely misspecified’ posterior inference on the volatility of a jump diffusion process

Bayesian analysis requires prior distributions for all model parameters, whether of interest or not. This can be a burden, for a number of reasons, especially when the nuisance parameters are high- or infinite-dimensional, so there is motivation to find a way around this without completely abandoning the Bayesian approach. Here we consider a general strategy of working with a purposely misspecified model to avoid dealing directly with nuisance parameters. We focus this investigation on an interesting and challenging problem of inference on the volatility of a jump diffusion process based on discrete observations. If we simply ignore the jumps, we can work out precisely the asymptotic behavior of the Bayesian posterior distribution based on the misspecified model. This result suggests some simple adjustments to correct for the effects of misspecification, and we demonstrate that a suitably corrected version of our purposely misspecified posterior leads to inference on the volatility that is asymptotically optimal.