Abstract

This work aims to make a comprehensive contribution in the area of Bayesian inference for discretely observed diffusion processes. Established approaches for likelihood-based inference invoke a numerical 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 upon specification of a sufficiently small step-size. Recent research (Ditlevsen and Samson, 2019) proposes the use of the strong 1.5 order scheme (Kloeden and Platen, 1992) and prove consistency and asymptotic normality for the Maximum Likelihood Estimator (MLE) in the setting of hypoelliptic SDEs for the common high-frequency observation regime. The numerical scheme is well understood to have a strong effect on the computational performance of data augmentation algorithms. However, the applicability of the strong 1.5 order scheme is restricted to SDEs whose diffusion coefficients satisfy a specific condition. Gloter and Yoshida (2021) suggest a simpler Gaussian scheme that is widely applicable for hypoelliptic SDEs and produce corresponding asymptotic results. Our work proposes a weak 2 order scheme which allows for general applicability in both elliptic and hypoelliptic models and provides superior computation performance. Thus, we recommend the scheme as a universal, effective tool for the time-discretisation of SDEs when applying data augmentation. Furthermore, following standard literature, we obtain full asymptotic results for the deduced MLE, which permits uncertainty quantification in the classical frequentist setting. This is a joint work with Alexandros Beskos and Matthew M. Graham.