  In this article we consider parametric Bayesian inference for stochastic
differential equations (SDE) driven by a pure-jump stable Levy process, which
is observed at high frequency. In most cases of practical interest, the
likelihood function is not available, so we use a quasi-likelihood and place an
associated prior on the unknown parameters. It is shown under regularity
conditions that there is a Bernstein-von Mises theorem associated to the
posterior. We then develop a Markov chain Monte Carlo (MCMC) algorithm for
Bayesian inference and assisted by our theoretical results, we show how to
scale Metropolis-Hastings proposals when the frequency of the data grows, in
order to prevent the acceptance ratio going to zero in the large data limit.
Our algorithm is presented on numerical examples that help to verify our
theoretical findings.
