We consider the problem of Bayesian estimation of static parameters associated to a partially and discretely observed diffusion process. We assume that the exact transition dynamics of the diffusion process are unavailable, even up-to an unbiased estimator and that one must time-discretize the diffusion process. In such scenarios it has been shown how one can introduce the multilevel Monte Carlo method to reduce the cost to compute posterior expected values of the parameters for a pre-specified mean square error (MSE). These afore-mentioned methods rely on upon the Euler-Maruyama discretization scheme which is well-known in numerical analysis to have slow convergence properties. We adapt stochastic Runge-Kutta (SRK) methods for Bayesian parameter estimation of static parameters for diffusions. This can be implemented in high-dimensions of the diffusion and seemingly under-appreciated in the uncertainty quantification and statistics fields. For a class of diffusions and SRK methods, we consider the estimation of the posterior expectation of the parameters. We prove that to achieve a MSE of $\mathcal{O}(\epsilon^2)$, for $\epsilon>0$ given, the associated work is $\mathcal{O}(\epsilon^{-2})$. Whilst the latter is achievable for the Milstein scheme, this method is often not applicable for diffusions in dimension larger than two. We also illustrate our methodology in several numerical examples.

翻译：本文研究与部分离散观测扩散过程相关的静态参数贝叶斯估计问题。假设扩散过程的精确转移动力学不可得（即使借助无偏估计量亦无法获得），且必须对该扩散过程进行时间离散化。已有研究表明，在此类场景中可引入多级蒙特卡洛方法，以降低为达到预设均方误差（MSE）而需计算的参数后验期望值成本。上述方法依赖于欧拉-丸山离散化格式，而该格式在数值分析领域公认具有收敛速度缓慢的特性。我们采用随机龙格-库塔（SRK）方法实现扩散过程静态参数的贝叶斯参数估计。该方法可推广至高维扩散场景，但在不确定性量化与统计学领域尚未得到充分重视。针对特定类别扩散过程与SRK方法，我们探讨了参数后验期望的估计问题。理论证明：为达到$\mathcal{O}(\epsilon^2)$的均方误差（给定$\epsilon>0$），相应计算复杂度为$\mathcal{O}(\epsilon^{-2})$。尽管米尔斯泰因格式亦可实现相同复杂度，但该格式通常不适用于维度大于二的扩散过程。此外，我们通过多个数值算例验证了所提方法的有效性。