# Estimating doubly stochastic Poisson process with affine intensities by Kalman filter

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## Abstract

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.

## Keywords

Doubly stochastic Poisson process Affine diffusion Kalman filter Order book## Mathematics Subject Classification

62M99 62P05## Notes

### Acknowledgments

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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