Statistical Papers

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

Estimating doubly stochastic Poisson process with affine intensities by Kalman filter

Regular Article
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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 

References

  1. 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
  2. Albanese C, Lawi S (2004) Laplace transforms for integrals of markov processes. Markov Process Rel Fields 11:677–724MathSciNetGoogle Scholar
  3. Basu S, Dassios A (2002) A cox process with log-normal intensity. Insur Math Econ 31:297–302MathSciNetCrossRefMATHGoogle Scholar
  4. Bielecki TR, Rutkowski M (2002) Credit risk: modeling, valuation and hedging. Springer, BerlinGoogle Scholar
  5. Bolder DJ (2001) Yield curve modelling at the bank of Canada. Bank of Canada working paper 2001–2015Google Scholar
  6. Bollerslev T, Wooldridge JM (1992) Quasi-maximum likelihood estimation and inference in dynamic models with time-varying covariances. Econ Rev 11:143–172MathSciNetCrossRefMATHGoogle Scholar
  7. Bouzas PR, Valderrama MJ, Aguilera AM (2002) Forecasting a class of doubly Poisson processes. Stat Pap 43:507–523MathSciNetCrossRefMATHGoogle Scholar
  8. 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 30:1021–1032CrossRefMATHGoogle Scholar
  9. 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
  10. Brémaud P (1972) Point processes and queues: martingale dynamics. Springer, New YorkGoogle Scholar
  11. Breusch TS (1979) Testing for autocorrelation in dynamic linear models. Aust Econ Pap 17:334–355CrossRefGoogle Scholar
  12. Byrd RH, Schnabel RB, Schultz GA (1987) A trust region algorithm for nonlinearly constrained optimization. SIAM J Numer Anal 24:1152–1170MathSciNetCrossRefMATHGoogle Scholar
  13. 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
  14. Cont R, Stoikov S, Talreja R (2010) A stochastic model for order book dynamics. Oper Res 10(3):549–563MathSciNetCrossRefGoogle Scholar
  15. Cox DR (1955) Some statistical methods connected with series of events. J R Stat Soc B 17:129–164MATHGoogle Scholar
  16. Cox J, Ingersoll J, Ross S (1985) A theory of the term structure of interest rates. Econometrica 53:385–408MathSciNetCrossRefMATHGoogle Scholar
  17. Dalal S, McIntosh A (1994) When to stop testing for large software systems with changing code. IEEE Trans Softw Eng 20:318–323CrossRefGoogle Scholar
  18. Daley DJ, Vere-Jones D (1988) An introduction to theory of point processes. Springer, New YorkMATHGoogle Scholar
  19. De Genaro A (2011) Cox processes with affine intensity. PhD. Thesis, Institute of Mathematics and Statistics-IME USP, Sao PauloGoogle Scholar
  20. 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
  21. 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
  22. Dassios A, Jang J (2012) A double shot-noise process and its application in insurance. J Math Syst Sci 2:82–93Google Scholar
  23. Duan J, Simonato J (1999) Estimating and testing exponential-affine term structure models by kalman filter. Rev Quant Financ Acc 13:111–135CrossRefGoogle Scholar
  24. Duffie D, Pan J, Singleton K (2010) Transform analysis and asset pricing for affine jump-diffusions. Econometrica 68(6):1343–1376MathSciNetCrossRefGoogle Scholar
  25. Duffie D, Filipović D, Schachermayer W (2003) Affine processes and applications in finance. Ann Appl Probab 13:984–1053MathSciNetCrossRefMATHGoogle Scholar
  26. Duffie D, Kan R (1996) A yield-factor model of interest rates. Math Financ 6:379–406CrossRefMATHGoogle Scholar
  27. Duffie D, Singleton K (1999) Modeling term structures defautable bonds. Rev Financ Stud 12:687–720CrossRefGoogle Scholar
  28. Dyrting S (2004) Evaluating the noncentral chi-square distribution for the Cox–Ingersoll–Ross process. Comput Econ 24:35–50CrossRefMATHGoogle Scholar
  29. Engle R, Russell J (1998) Autoregressive conditional duration: a new model for irregularly spaced transaction data. Econometrica 66:1127–1162MathSciNetCrossRefMATHGoogle Scholar
  30. Engle R, Russell J (2000) The econometrics of ultra-high-frequency data. Econometrica 68–1:1–22CrossRefGoogle Scholar
  31. Feller W (1951) Two singular diffusion problems. Ann Math 54:173–182MathSciNetCrossRefMATHGoogle Scholar
  32. Gail M, Santner T, Brown C (1980) An analysis of comparative carcinogenesis experiments based on multiple times to tumor. Biometrics 36:255–266MathSciNetCrossRefMATHGoogle Scholar
  33. 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
  34. Godfrey LG (1978) Testing against general autoregressive and moving average error models when the regressors include lagged dependent variables. Econometrica 46:1293–1302MathSciNetCrossRefMATHGoogle Scholar
  35. Grandell J (1976) Doubly stochastic process, 1st edn. Springer, New YorkGoogle Scholar
  36. Grandell J (1991) Aspects of risk theory. Springer, New YorkCrossRefMATHGoogle Scholar
  37. Grasselli M, Tebaldi C (2008) Solvable affine term structure models. Math Financ 18:135–153MathSciNetCrossRefMATHGoogle Scholar
  38. Hamilton J (1994) Time series analysis. Princeton University Press, PrincetonMATHGoogle Scholar
  39. Harvey A (1989) Forecasting, structural time series models and the Kalman Filter. Cambridge University Press, CambridgeGoogle Scholar
  40. Johnson N, Kotz S (1970) Distributions in statistics: continuous univariate distributions, vol 2. Wiley, New YorkMATHGoogle Scholar
  41. Kallenberg O (1986) Random measures, 4th edn. Academic Press, LondonGoogle Scholar
  42. Karatzas I, Shreve S (1991) Brownian motion and stochastic calculus, 2nd edn. Springer, New YorkMATHGoogle Scholar
  43. Karlin S, Taylor H (1981) A second course in stochastic process. Academic Press, New YorkGoogle Scholar
  44. Kozachenko YuV, Pogorilyak OO (2008) A method of modelling log Gaussian Cox process. Theory Probab Math Stat 77:91–105MathSciNetCrossRefGoogle Scholar
  45. Lando D (1998) On cox processes and credit risky securities. Rev Deriv Res 2:99–120MATHGoogle Scholar
  46. Ljung GM, Box G (1978) On a measure of lack of fit in time series models. Biometrika 62–2:297–303CrossRefGoogle Scholar
  47. 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
  48. Seal H (1983) The Poisson process: its failure in risk theory. Insur Math Econ 2–4:287–288. London: Croom Helm, 1979Google Scholar
  49. Sankaran M (1963) Approximations to the non-central chi-square distribution. Biometrika 50:199–204MathSciNetCrossRefMATHGoogle Scholar
  50. Snyder D, Miller M (1991) Random point processes in time and space, 2nd edn. Springer, New YorkCrossRefMATHGoogle Scholar
  51. Vasicek O (1977) An equilibrium characterization of the term structure. J Financ Econ 5:177–188CrossRefGoogle Scholar
  52. Vuong QH (1989) Likelihood ratio tests for model selection and non-nested hypothesis. Econometrica 57:307–333MathSciNetCrossRefMATHGoogle Scholar
  53. 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
  54. Zhang T, Kou S (2010) Nonparametric inference of doubly stochastic Poisson process data via kernel method. Ann Appl Stat 4:1913–1941MathSciNetCrossRefMATHGoogle Scholar

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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