Modelling random dynamical systems in continuous time, diffusion processes are a powerful tool in many areas of science. Model parameters can be estimated from time-discretely observed 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 accelerate it, but the specific implementation and benefit remain unclear. Hence, we investigate the utilization and usefulness of higher-order approximations in the example of the Milstein scheme. Our study demonstrates that the MCMC methods based on the Milstein approximation yield good estimation results. However, they are computationally more expensive and can be applied to multidimensional processes only with impractical restrictions. Moreover, the combination of the Milstein approximation and the well-known modified bridge proposal introduces additional numerical challenges.

在连续时间内对随机动力系统进行建模，扩散过程是许多科学领域的强大工具。可以使用引入辅助数据的马尔可夫链蒙特卡罗 (MCMC) 方法从时间离散观察过程中估计模型参数。这些方法通常以数字方式近似过程的转变密度，既用于计算后验密度又用于提出辅助数据。在这里，Euler-Maruyama 方案是标准的近似技术。然而，MCMC 方法的计算成本较高。使用高阶近似可能会加速它，但具体实现和好处仍不清楚。因此，我们在米尔斯坦方案的例子中研究了高阶近似的利用和有用性。我们的研究表明，基于 Milstein 近似的 MCMC 方法产生了良好的估计结果。然而，它们的计算成本更高，并且只能在不切实际的限制下应用于多维过程。此外，米尔斯坦近似和著名的修改桥方案的结合引入了额外的数值挑战。