Numerical Schemes for Effective Calibration of Elliptic and Hypo-elliptic Diffusions

This work aims at making a comprehensive contribution in the general area of parametric inference for partially observed diffusion processes. Established approaches for likelihood-based estimation invoke a numerical time-discretisation scheme for the approximation of the (typically intractable) transition dynamics of the Stochastic Differential Equation (SDE) model over finite time periods. The scheme is applied for a step-size that is either a user-selected tuning parameter or determined by the data. Recent research has highlighted the critical effect of the choice of numerical scheme on the behaviour of derived parameter estimates in the setting of hypo-elliptic SDEs. In brief, in our work, first, we develop two weak second order `sampling schemes' (to cover both the hypo-elliptic and elliptic SDE classes) and generate accompanying `transition density schemes' of the SDE (i.e., approximations of the SDE transition density). Then, we produce a collection of analytic results, providing a complete theoretical framework that solidifies the proposed schemes and showcases advantages from their incorporation within SDE calibration methods. We present numerical results from carrying out classical or Bayesian inference, for both elliptic and hypo-elliptic SDE models.