  Jump diffusion processes are widely used to model asset prices over time,
mainly for their ability to capture complex discontinuous behavior, but
inference on the model parameters remains a challenge. Here our goal is
posterior inference on the volatility coefficient of the diffusion part of the
process based on discrete samples. A Bayesian approach requires specification
of a model for the jump part of the process, prior distributions for the
corresponding parameters, and computation of the joint posterior. Since the
volatility coefficient is our only interest, it would be desirable to avoid the
modeling and computational costs associated with the jump part of the process.
Towards this, we consider a {\em purposely misspecified model} that ignores the
jump part entirely. We work out precisely the asymptotic behavior of the
Bayesian posterior under the misspecified model, propose some simple
modifications to correct for the effects of misspecification, and demonstrate
that our modified posterior inference on the volatility is efficient in the
sense that its asymptotic variance equals the no-jumps model Cram\'er--Rao
bound.
