In this paper we consider the filtering of partially observed multi-dimensional diffusion processes that are observed regularly at discrete times. This is a challenging problem which requires the use of advanced numerical schemes based upon time-discretization of the diffusion process and then the application of particle filters. Perhaps the state-of-the-art method for moderate dimensional problems is the multilevel particle filter of \cite{mlpf}. This is a method that combines multilevel Monte Carlo and particle filters. The approach in that article is based intrinsically upon an Euler discretization method. We develop a new particle filter based upon the antithetic truncated Milstein scheme of \cite{ml_anti}. We show that for a class of diffusion problems, for $\epsilon>0$ given, that the cost to produce a mean square error (MSE) in estimation of the filter, of $\mathcal{O}(\epsilon^2)$ is $\mathcal{O}(\epsilon^{-2}\log(\epsilon)^2)$. In the case of multidimensional diffusions with non-constant diffusion coefficient, the method of \cite{mlpf} has a cost of $\mathcal{O}(\epsilon^{-2.5})$ to achieve the same MSE. We support our theory with numerical results in several examples.

翻译：本文研究了部分观测的多维扩散过程在离散时间点上的滤波问题。这是一个具有挑战性的问题，需要采用基于扩散过程时间离散化的先进数值方案，并随后应用粒子滤波器。目前，针对中等维度问题的最先进方法可能是《多级粒子滤波器》(参考文献 mlpf)。该方法结合了多级蒙特卡洛与粒子滤波器，其核心基于欧拉离散化方案。我们基于《对偶截断米尔斯坦方案》(参考文献 ml_anti) 开发了一种新的粒子滤波器。我们证明，对于一类扩散问题，给定 $\epsilon>0$，实现滤波器估计均方误差 (MSE) 为 $\mathcal{O}(\epsilon^2)$ 的计算成本为 $\mathcal{O}(\epsilon^{-2}\log(\epsilon)^2)$。而对于具有非常数扩散系数的多维扩散过程，采用文献 mlpf 中的方法达到相同 MSE 的成本为 $\mathcal{O}(\epsilon^{-2.5})$。我们通过多个数值算例验证了理论结果。