In this article we consider the filtering problem associated to partially observed diffusions, with observations following a marked point process. In the model, the data form a point process with observation times that have its intensity driven by a diffusion, with the associated marks also depending upon the diffusion process. We assume that one must resort to time-discretizing the diffusion process and develop particle and multilevel particle filters to recursively approximate the filter. In particular, we prove that our multilevel particle filter can achieve a mean square error (MSE) of $\mathcal{O}(\epsilon^2)$ ($\epsilon>0$ and arbitrary) with a cost of $\mathcal{O}(\epsilon^{-2.5})$ versus using a particle filter which has a cost of $\mathcal{O}(\epsilon^{-3})$ to achieve the same MSE. We then show how this methodology can be extended to give unbiased (that is with no time-discretization error) estimators of the filter, which are proved to have finite variance and with high-probability have finite cost. Finally, we extend our methodology to the problem of online static-parameter estimation.

翻译：本文考虑与部分观测扩散过程相关的滤波问题，其中观测值遵循标记点过程。在该模型中，数据构成一个点过程，其观测时间由扩散过程驱动的强度决定，同时相关标记也依赖于扩散过程。我们假设必须对扩散过程进行时间离散化，并开发粒子滤波器和多水平粒子滤波器以递归逼近滤波器。特别地，我们证明，与实现相同均方误差（MSE）成本为$\mathcal{O}(\epsilon^{-3})$的粒子滤波器相比，我们的多水平粒子滤波器能以$\mathcal{O}(\epsilon^{-2.5})$的成本实现均方误差（MSE）$\mathcal{O}(\epsilon^2)$（其中$\epsilon>0$且任意）。随后，我们展示如何将该方法扩展为滤波器的无偏估计量（即无时间离散化误差），并证明其具有有限方差且以高概率具有有限成本。最后，我们将方法推广至在线静态参数估计问题。