  Statistical inference for discretely observed jump-diffusion processes is a
complex problem which motivates new methodological challenges. Thus existing
approaches invariably resort to time-discretisations which inevitably lead to
approximations in inference. In this paper, we give the first general
collection of methodologies for exact (in this context meaning
discretisation-free) likelihood-based inference for discretely observed finite
activity jump-diffusions. The only sources of error involved are Monte Carlo
error and convergence of EM or MCMC algorithms. We shall introduce both
frequentist and Bayesian approaches, illustrating the methodology through
simulated and real examples.
