# 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.

## References

- Akaike H (1973) Information theory and an extension of the maximum likelihood principle. In: Proceedings of the 2nd international symposium on information theory, pp 267–281Google Scholar
- Albanese C, Lawi S (2004) Laplace transforms for integrals of markov processes. Markov Process Rel Fields 11:677–724MathSciNetGoogle Scholar
- Basu S, Dassios A (2002) A cox process with log-normal intensity. Insur Math Econ 31:297–302MathSciNetCrossRefzbMATHGoogle Scholar
- Bielecki TR, Rutkowski M (2002) Credit risk: modeling, valuation and hedging. Springer, BerlinGoogle Scholar
- Bolder DJ (2001) Yield curve modelling at the bank of Canada. Bank of Canada working paper 2001–2015Google Scholar
- Bollerslev T, Wooldridge JM (1992) Quasi-maximum likelihood estimation and inference in dynamic models with time-varying covariances. Econ Rev 11:143–172MathSciNetCrossRefzbMATHGoogle Scholar
- Bouzas PR, Valderrama MJ, Aguilera AM (2002) Forecasting a class of doubly Poisson processes. Stat Pap 43:507–523MathSciNetCrossRefzbMATHGoogle Scholar
- Bouzas PR, Valderrama MJ, Aguilera AM (2006) On the characteristic functional of a doubly stochastic poisson process: application to a narrow-band process. Appl Math Model 30:1021–1032CrossRefzbMATHGoogle Scholar
- Bouzas PR, Ruiz-Fuentes N, Mantilla A, Valderrama MJ, Aguilera AM (2010) A Cox model for radioactive counting measure: inference on the intensity process. Chemometr Intell Lab 103:116–121CrossRefGoogle Scholar
- Brémaud P (1972) Point processes and queues: martingale dynamics. Springer, New YorkGoogle Scholar
- Breusch TS (1979) Testing for autocorrelation in dynamic linear models. Aust Econ Pap 17:334–355CrossRefGoogle Scholar
- Byrd RH, Schnabel RB, Schultz GA (1987) A trust region algorithm for nonlinearly constrained optimization. SIAM J Numer Anal 24:1152–1170MathSciNetCrossRefzbMATHGoogle Scholar
- Chen R-R, Scott L (2003) Multi-factor Cox–Ingersoll–Ross models of the term structure: estimates and tests from a Kalman filter model. J Real Estate Financ Econ 27:143–172CrossRefGoogle Scholar
- Cont R, Stoikov S, Talreja R (2010) A stochastic model for order book dynamics. Oper Res 10(3):549–563MathSciNetCrossRefGoogle Scholar
- Cox DR (1955) Some statistical methods connected with series of events. J R Stat Soc B 17:129–164zbMATHGoogle Scholar
- Cox J, Ingersoll J, Ross S (1985) A theory of the term structure of interest rates. Econometrica 53:385–408MathSciNetCrossRefzbMATHGoogle Scholar
- Dalal S, McIntosh A (1994) When to stop testing for large software systems with changing code. IEEE Trans Softw Eng 20:318–323CrossRefGoogle Scholar
- Daley DJ, Vere-Jones D (1988) An introduction to theory of point processes. Springer, New YorkzbMATHGoogle Scholar
- De Genaro A (2011) Cox processes with affine intensity. PhD. Thesis, Institute of Mathematics and Statistics-IME USP, Sao PauloGoogle Scholar
- Dassios A, Jang J (2003) Pricing of castrophe reinsurance and derivatives using the Cox process with shot noise intensity. Financ Stoch 7:73–95MathSciNetCrossRefGoogle Scholar
- Dassios A, Jang J (2008) The distribution of the interval between events of a cox process with shot noise intensity. J Appl Math Stoch Anal 2008:1–14Google Scholar
- Dassios A, Jang J (2012) A double shot-noise process and its application in insurance. J Math Syst Sci 2:82–93Google Scholar
- Duan J, Simonato J (1999) Estimating and testing exponential-affine term structure models by kalman filter. Rev Quant Financ Acc 13:111–135CrossRefGoogle Scholar
- Duffie D, Pan J, Singleton K (2010) Transform analysis and asset pricing for affine jump-diffusions. Econometrica 68(6):1343–1376MathSciNetCrossRefGoogle Scholar
- Duffie D, Filipović D, Schachermayer W (2003) Affine processes and applications in finance. Ann Appl Probab 13:984–1053MathSciNetCrossRefzbMATHGoogle Scholar
- Duffie D, Kan R (1996) A yield-factor model of interest rates. Math Financ 6:379–406CrossRefzbMATHGoogle Scholar
- Duffie D, Singleton K (1999) Modeling term structures defautable bonds. Rev Financ Stud 12:687–720CrossRefGoogle Scholar
- Dyrting S (2004) Evaluating the noncentral chi-square distribution for the Cox–Ingersoll–Ross process. Comput Econ 24:35–50CrossRefzbMATHGoogle Scholar
- Engle R, Russell J (1998) Autoregressive conditional duration: a new model for irregularly spaced transaction data. Econometrica 66:1127–1162MathSciNetCrossRefzbMATHGoogle Scholar
- Engle R, Russell J (2000) The econometrics of ultra-high-frequency data. Econometrica 68–1:1–22CrossRefGoogle Scholar
- Feller W (1951) Two singular diffusion problems. Ann Math 54:173–182MathSciNetCrossRefzbMATHGoogle Scholar
- Gail M, Santner T, Brown C (1980) An analysis of comparative carcinogenesis experiments based on multiple times to tumor. Biometrics 36:255–266MathSciNetCrossRefzbMATHGoogle Scholar
- Geye A, Pichler S (1999) A state-space approach to estimate and test multifactor Cox–Ingersoll–Ross models of the term structure of interest rates. J Financ Res 22:107–130CrossRefGoogle Scholar
- Godfrey LG (1978) Testing against general autoregressive and moving average error models when the regressors include lagged dependent variables. Econometrica 46:1293–1302MathSciNetCrossRefzbMATHGoogle Scholar
- Grandell J (1976) Doubly stochastic process, 1st edn. Springer, New YorkGoogle Scholar
- Grandell J (1991) Aspects of risk theory. Springer, New YorkCrossRefzbMATHGoogle Scholar
- Grasselli M, Tebaldi C (2008) Solvable affine term structure models. Math Financ 18:135–153MathSciNetCrossRefzbMATHGoogle Scholar
- Hamilton J (1994) Time series analysis. Princeton University Press, PrincetonzbMATHGoogle Scholar
- Harvey A (1989) Forecasting, structural time series models and the Kalman Filter. Cambridge University Press, CambridgeGoogle Scholar
- Johnson N, Kotz S (1970) Distributions in statistics: continuous univariate distributions, vol 2. Wiley, New YorkzbMATHGoogle Scholar
- Kallenberg O (1986) Random measures, 4th edn. Academic Press, LondonGoogle Scholar
- Karatzas I, Shreve S (1991) Brownian motion and stochastic calculus, 2nd edn. Springer, New YorkzbMATHGoogle Scholar
- Karlin S, Taylor H (1981) A second course in stochastic process. Academic Press, New YorkGoogle Scholar
- Kozachenko YuV, Pogorilyak OO (2008) A method of modelling log Gaussian Cox process. Theory Probab Math Stat 77:91–105MathSciNetCrossRefGoogle Scholar
- Lando D (1998) On cox processes and credit risky securities. Rev Deriv Res 2:99–120zbMATHGoogle Scholar
- Ljung GM, Box G (1978) On a measure of lack of fit in time series models. Biometrika 62–2:297–303CrossRefGoogle Scholar
- Minozzo M, Centanni S (2012) Monte Carlo likelihood inference for marked doubly stochastic Poisson processes with intensity driven by marked point processes. Working Paper Series, Dept. Economics, University of VeronaGoogle Scholar
- Seal H (1983) The Poisson process: its failure in risk theory. Insur Math Econ 2–4:287–288. London: Croom Helm, 1979Google Scholar
- Sankaran M (1963) Approximations to the non-central chi-square distribution. Biometrika 50:199–204MathSciNetCrossRefzbMATHGoogle Scholar
- Snyder D, Miller M (1991) Random point processes in time and space, 2nd edn. Springer, New YorkCrossRefzbMATHGoogle Scholar
- Vasicek O (1977) An equilibrium characterization of the term structure. J Financ Econ 5:177–188CrossRefGoogle Scholar
- Vuong QH (1989) Likelihood ratio tests for model selection and non-nested hypothesis. Econometrica 57:307–333MathSciNetCrossRefzbMATHGoogle Scholar
- Wei G, Clifford P, Feng J (2002) Population death sequences and cox processes driven by interacting Feller diffusions. J Phys A Math Gen 35:9–31Google Scholar
- Zhang T, Kou S (2010) Nonparametric inference of doubly stochastic Poisson process data via kernel method. Ann Appl Stat 4:1913–1941MathSciNetCrossRefzbMATHGoogle Scholar