Unbiased and multilevel methods for a class of diffusions partially observed via marked point processes

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 \(\mathcal {O}(\epsilon ^2)\) $\mathcal{O}({ϵ}^2)$ ( \(\epsilon >0\) $ϵ>0$ and arbitrary) with a cost of \(\mathcal {O}(\epsilon ^{-2.5})\) $\mathcal{O}({ϵ}^{-2.5})$ versus using a particle filter which has a cost of \(\mathcal {O}(\epsilon ^{-3})\) $\mathcal{O}({ϵ}^{-3})$ 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.