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Statistical Inference for Stochastic Processes

, Volume 18, Issue 3, pp 293–313 | Cite as

Stability of the filter with Poisson observations

  • Zhiqiang Li
  • Jie XiongEmail author
Article
  • 136 Downloads

Abstract

The short interest rate process is modeled by a diffusion process \(X(t)\). With the counting process observations, a filtering problem is formulated and its exponential stability is derived when the process \(X(t)\) is asymptotically stationary.

Keywords

Diffusion process Truncated filter Stability 

Mathematics Subject Classification

Primary 93E11 Secondary: 60J25 62M20 

Notes

Acknowledgments

This research supported partially by FDCT 076/2012/A3. The authors would like to thank the anonymous reviewer for the constructive suggestions and comments which improved this paper substantially.

References

  1. Atar R (1998) Exponential stability for nonlinear filtering of diffusion processes in non-compact domain. Ann Probab 26:1552–1574zbMATHMathSciNetCrossRefGoogle Scholar
  2. Atar R, Zeitouni O (1997a) Exponential stability for nonlinear filtering. Ann Inst Henri Poincare Probab Stat 33:697–725Google Scholar
  3. Atar R, Zeitouni O (1997b) Lyapunov exponents for finite state nonlinear filtering. SIAM J Control Optim 35:36–55Google Scholar
  4. Baxendale P, Chigansky P, Liptser R (2004) Asymptotic stability of the Wonham filter: ergodic and nonergodic signals. SIAM J Control Optim 43(2):643–669zbMATHMathSciNetCrossRefGoogle Scholar
  5. Budhiraja A, Kushner HJ (1998) Robustness of nonlinear filters over the infinite time interval. SIAM J Control Optim 36:1618–1637zbMATHMathSciNetCrossRefGoogle Scholar
  6. Budhiraja A, Kushner HJ (1999) Approximation and limit results for nonlinear filters over an infinite time interval. SIAM J Control Optim 37(6):1946–1979zbMATHMathSciNetCrossRefGoogle Scholar
  7. Budhiraja A, Kushner HJ (2000) Approximation and limit results for nonlinear filters over an infinite time interval. II. Random sampling algorithms. SIAM J Control Optim 38(6):1874–1908zbMATHMathSciNetCrossRefGoogle Scholar
  8. Budhiraja A, Kushner HJ (2001) Monte Carlo algorithms and asymptotic problems in nonlinear filtering. In: Hida T et al (eds) Stochastics in finite/infinite dimension. Trends in mathematics series. Birkhäuser, Boston, pp 59–87Google Scholar
  9. Budhiraja A, Ocone DL (1997) Exponential stability of discrete time filters without signal ergodicity. Syst Control Lett 30:185–193zbMATHMathSciNetCrossRefGoogle Scholar
  10. Budhiraja A, Ocone DL (1999) Exponential stability in discrete time filtering for non-ergodic signals. Stoch Process Appl 82:245–257zbMATHMathSciNetCrossRefGoogle Scholar
  11. Chigansky P (2006a) An ergodic theorem for filtering with applications to stability. Syst Control Lett 55(11):908–917Google Scholar
  12. Chigansky P (2006b) Stability of the nonlinear filter for slowly switching Markov chains. Stoch Process Appl 116(8):1185–1194Google Scholar
  13. Chigansky P, Liptser R (2004) Stability of nonlinear filters in nonmixing case. Ann Appl Probab 14(4):2038–2056zbMATHMathSciNetCrossRefGoogle Scholar
  14. Cox JC, Ingersoll JE, Ross SA (1985) A theory of the term structure of interest rates. Econometrica 53:385–407zbMATHMathSciNetCrossRefGoogle Scholar
  15. Da Prato G, Fuhrman M, Malliavin P (1995) Asymptotic ergodicity for the Zakai filtering equation. C R Acad Sci Paris Sér I Math I 321:613–616zbMATHGoogle Scholar
  16. Del Moral P, Guionnet A (2001) On the stability of interacting processes with applications to filtering and genetic algorithms. Ann Inst Henri Poincare Probab Stat 37(2):155–194zbMATHCrossRefGoogle Scholar
  17. Del Moral P, Miclo L (2002) On the stability of nonlinear Feynman–Kac semigroups. Ann Fac Sci Toulouse Math (6) 11(2):135–175zbMATHMathSciNetCrossRefGoogle Scholar
  18. Delyon B, Zeitouni O (1991) Lyapunov exponents for filtering problem. In: Davis MHA, Elliot RJ (eds) Applied stochastic analysis. Gordon & Breach, New York, pp 511–521Google Scholar
  19. Di Masi GB, Stettner L (2005) Ergodicity of hidden Markov models. Math Control Signals Syst 17(4):269–296zbMATHCrossRefGoogle Scholar
  20. Hasbrouck J (1996) Modeling market microstructure time series. In: Maddala GS, Rao CR (eds) Handbook of statistics. North-Holland, Amsterdam, pp 647–692Google Scholar
  21. Hasbrouck J (2002) Stalking the “efficient price” in market microstructure specifications: an overview. J Financ Mark 5:329–339CrossRefGoogle Scholar
  22. Hopf E (1963) An inequality for positive linear integral operators. J Math Mech 12(5):683–692zbMATHMathSciNetGoogle Scholar
  23. Kouritzin M, Zeng Y (2005) Bayesian model selection via filtering for a class of micromovement models of asset price. Int J Theor Appl Finance 8:97–121zbMATHMathSciNetCrossRefGoogle Scholar
  24. Krasnosel’hskii MA, Lifshits EA, Sobolev VA (1989) Positive linear systems. The method of positive operators. Heldermann, BerlinGoogle Scholar
  25. Lee K, Zeng Y (2010) Risk minimization for a filtering micromovement model of asset price. Appl Math Finance 17(2):177199MathSciNetCrossRefGoogle Scholar
  26. Le Gland F, Mevel L (2000) Exponential forgetting and geometric ergodicity in hidden Markov models. Math Control Signals Syst 13:63–93zbMATHCrossRefGoogle Scholar
  27. Le Gland F, Oudjane N (2003) A robustification approach to stability and to uniform particle approximation of nonlinear filters: the example of pseudo-mixing signals. Stoch Process Appl 106(2):279–316CrossRefGoogle Scholar
  28. Le Gland F, Oudjane N (2004) Stability and uniform approximation of nonlinear filters using the Hilbert metric and application to particle filters. Ann Appl Probab 14(1):144–187zbMATHMathSciNetCrossRefGoogle Scholar
  29. Morters P, Peres Y (2010) Brownian motion. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  30. Oudjane N, Rubenthaler S (2005) Stability and uniform particle approximation of nonlinear filters in case of non ergodic signals. Stoch Anal Appl 23(3):421–448zbMATHMathSciNetCrossRefGoogle Scholar
  31. Papavasiliou A (2006) Parameter estimation and asymptotic stability in stochastic filtering. Stoch Process Appl 116(7):1048–1065zbMATHMathSciNetCrossRefGoogle Scholar
  32. Rendleman R, Bartte B (1980) The pricing of options on debt securities. J Financ Quant Anal 15:11–24CrossRefGoogle Scholar
  33. Tadić VB, Doucet A (2005) Exponential forgetting and geometric ergodicity for optimal filtering in general state-space models. Stoch Process Appl 115(8):1408–1436zbMATHCrossRefGoogle Scholar
  34. Vasicek O (1977) An equilibrium characterisation of the term structure. J Financ Econ 5:177–188CrossRefGoogle Scholar
  35. Xiong J (2008) An introduction to stochastic filtering theory. Oxford University Press, OxfordzbMATHGoogle Scholar
  36. Xiong J, Zeng Y (2011) A branching particle approximation to a filtering micromovement model of asset price. Stat Inference Stoch Process 14(2):111–140zbMATHMathSciNetCrossRefGoogle Scholar
  37. Xiong J, Zeng Y (2014) Mean-variance porfolio selection for a filtering point process model of asset price. Working Paper, University of TennesseeGoogle Scholar
  38. Zeng Y (2003) A partialy observed model for micromovement of asset prices with Bayes estimation via filtering. Math Finance 13:414–444CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TennesseeKnoxvilleUSA
  2. 2.Department of Mathematics, Faculty of Science and TechnologyUniversity of MacauTaipaMacau, China

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