A powerful tool in many areas of science, diffusion processes model random dynamical systems in continuous time. Parameters can be estimated from time-discretely observed diffusion processes using Markov chain Monte Carlo (MCMC) methods that introduce auxiliary data. These methods typically approximate the transition densities of the process numerically, both for calculating the posterior densities and proposing auxiliary data. Here, the Euler-Maruyama scheme is the standard approximation technique. However, the MCMC method is computationally expensive. Using higher-order approximations may speed it up. Yet, the specific such implementation and benefit remain unclear. Hence, we investigate the utilisation and usefulness of higher-order approximations on the example of the Milstein scheme. Our study shows that the combination of the Milstein approximation and the well-known modified bridge proposal yields good estimation results. However, this proceeding is computationally more expensive, introduces additional numerical challenges and can be applied to multidimensional processes only with impractical restrictions.
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