Exact Monte Carlo likelihood-based inference for jump-diffusion processes

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.