2023/11/16 by Alvarez, Miguel, Jasra, Ajay, Ruzayqat, Hamza
#60G35 #60G55 #62F30 #62M20 #Computation (stat.CO) #FOS: Computer and information sciences #FOS: Mathematics #Methodology (stat.ME) #Numerical Analysis (math.NA)
paper · doi:10.48550/arxiv.2311.09875
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 O(ε2) (ε>0 and arbitrary) with a cost of O(ε-2.5) versus using a particle filter which has a cost of 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.