Statistical Papers

, Volume 56, Issue 3, pp 723–748 | Cite as

Estimating doubly stochastic Poisson process with affine intensities by Kalman filter

  • Alan De GenaroEmail author
  • Adilson Simonis
Regular Article


This paper proposes a Kalman filter formulation for parameter estimation of doubly stochastic Poisson processes (DSPP) with stochastic affine intensities. To achieve this aim, an analytical expression for the probability distribution functions of the corresponding DSPP for any intensity from the class of affine diffusions is obtained. More detailed results are provided for one- and two-factor Feller and Ornstein–Uhlenbeck diffusions. A Monte Carlo study indicates that the proposed method is a reliable procedure for moderate sample sizes. An empirical analysis of one- and two-factor Feller and Ornstein–Uhlenbeck models is carried out using high frequency transaction data.


Doubly stochastic Poisson process Affine diffusion  Kalman filter Order book 

Mathematics Subject Classification

62M99 62P05 



Alan De Genaro would like to thank Marco Avellaneda, Jorge Zubelli, Cristiano Fernandes, Julio Stern, Peter Carr, Cris Rogers, Jean Pierre Fouque and seminar participants at NYU-Courant, IMPA, FEA/USP, SUNY—Stony Brook for helpful comments. We also thank the three reviewers for their thorough review and highly appreciate the comments and suggestions, which significantly contributed to improving the quality of this paper. A special thank is due to Yuri Suhov for his invaluable suggestions.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of Economics, University of Sao Paulo and SecuritiesCommodities and Futures Exchange, BM&FBOVESPASão PauloBrazil
  2. 2.Institute of Mathematics and StatisticsUniversity of Sao PauloSão PauloBrazil

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