Applied Mathematics & Optimization

, Volume 67, Issue 3, pp 323–351

Filtering with Marked Point Process Observations via Poisson Chaos Expansion


DOI: 10.1007/s00245-012-9189-6

Cite this article as:
Sun, W., Zeng, Y. & Zhang, S. Appl Math Optim (2013) 67: 323. doi:10.1007/s00245-012-9189-6


We study a general filtering problem with marked point process observations. The motivation comes from modeling financial ultra-high frequency data. First, we rigorously derive the unnormalized filtering equation with marked point process observations under mild assumptions, especially relaxing the bounded condition of stochastic intensity. Then, we derive the Poisson chaos expansion for the unnormalized filter. Based on the chaos expansion, we establish the uniqueness of solutions of the unnormalized filtering equation. Moreover, we derive the Poisson chaos expansion for the unnormalized filter density under additional conditions. To explore the computational advantage, we further construct a new consistent recursive numerical scheme based on the truncation of the chaos density expansion for a simple case. The new algorithm divides the computations into those containing solely system coefficients and those including the observations, and assign the former off-line.


Poisson chaos expansionFilteringMarked point processesRandom measuresUltra-high frequency data

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsConcordia UniversityMontrealCanada
  2. 2.Department of Mathematics and StatisticsUniversity of Missouri at Kansas CityKansas CityUSA