General smoothing formulas for Markov-modulated poisson observations

IEEE Transactions on Automatic Control

Volumn 50, Issue 8, 2005, Pages 1123-1134

  Elliott, Robert J ^a   Malcolm, W Paul ^b  

^a UNIVERSITY OF CALGARY   (Canada)

^b CSIRO   (Australia)

Author keywords

Filtering; Martingales; Poisson processes; Reference probability; Smoothing

Indexed keywords

ALGORITHMS; COMPUTER SIMULATION; ESTIMATION; INTEGRAL EQUATIONS; LINEAR EQUATIONS; MATHEMATICAL MODELS; MATHEMATICAL TRANSFORMATIONS; POISSON EQUATION; PROBABILITY; SIGNAL FILTERING AND PREDICTION; SIGNAL PROCESSING; THEOREM PROVING;

LEBESGUE-STIELTJES STOCHASTIC INTEGRALS; MARKOV-MODULATED POISSON OBSERVATION; MARTINGALES; POISSON PROCESSES; RADON-NIKODYM DERIVATIVES; REFERENCE PROBABILITY; SMOOTHING FORMULAS; STOCHASTIC INTEGRAL EQUATIONS;

MARKOV PROCESSES;

EID: 26244458556     PISSN: 00189286     EISSN: None     Source Type: Journal    
DOI: 10.1109/TAC.2005.852565     Document Type: Article

References (31)

1. P. Billingsley, Convergence of Probability Measures, 2nd ed. New York: Wiley, 1999.
2. P. Bremaud, Point Processes and Queues, Martingale Dynamics, ser. Springer Series in Statistics. New York: Springer-Verlag, 1981.
3. J. M. C. Clark, "The design of robust approximations to the stochastic differential equations for nonlinear filtering" in Communications Systems and Random Process Theory, J. K. Skwirzynski, Ed. Amsterdam, The Netherlands: Sijthoff and Noorhoff, 1978, pp. 721-734.
4. D. Clements and B. D. O. Anderson, "A nonlinear fixed-lag smoother for finite-state markov processes" IEEE Trans. Inf. Theory, vol. 21, no. 4, pp. 446-452, Jul. 1975.
5. M. H. A. Davis, "On a multiplicative functional transformation arising in nonlinear filtering theory" Zeitschrift fuer Wahrscheinlichkeitstheorie venvandete Gebiete, vol. 54, pp. 125-139, 1980.
6. R. J. Elliott, Stochastic Calculus and Its Applications. New York: Springer-Verlag, 1982.
7. R. J. Elliott, "New finite dimensional filters and smoothers for noisily observed markov chains" IEEE Trans. Inf. Theory, vol. 39, no. 1, pp. 265-271, Jan. 1993.
8. R. J. Elliott, L. Aggoun, and J. B. Moore, Hidden Markov Models Estimation and Control, ser. Applications of Mathematics. New York: Springer-Verlag, 1995.
9. G. Golub and C. van Loan, Matrix Computations. Baltimore, MD: John Hopkins Univ. Press. 1996.
10. R. J. Elliott, "Filtering and control for point process observations" in Recent Advances in Stochastic Analysis, J. Barras and V. Mirelli, Eds. New York: Springer-Verlag, 1990, pp. 1-26.
11. R. J. Elliott and W. P. Malcolm, "Robust smoother dynamics for poisson processes driven by an itô diffusion" in Proc. IEEE Conf. Decision and Control, vol. 1, Orlando, FL, Dec. 2001, pp. 376-381.
12. J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes, ser. Grundlehren der Mathematischen Wissenschaften. Berlin, Germany: Springer-Verlag, 1997.
13. M. R. James, V. Krishnamurthy, and F. L. Gland, "Time discretization for continuous time filters and smoothers for hmm parameter estimation" IEEE Trans. Inf. Theory, vol. 42, no. 2, pp. 596-605, Mar. 1996.
14. W. Kliemann, W. Koch, and F. Marchetti, "On the unnormalized solution of the filtering problem with counting process observations" IEEE Trans. Inf. Theory, vol. 36, no. 6, pp. 1415-1425, Nov. 1990.
15. P. E. Kloeden and E. Platen. Numerical Solution of Stochastic Differential Equations, ser. Applications of Mathematics. New York: Springer-Verlag, 1995.
16. T. Lindvall, "Weak convergence of probability measures and random functions in the function space d[0, inf)" J. Appl. Probab., no. 10, pp. 109-121, Nov. 1973.
17. W. P. Malcolm, Robust Filtering and Estimation With Poisson Observations. Canberra, Australia: Australian National Univ., 1999.
18. W. P. Malcolm and R. J. Elliott, "A general smoothing equation for poisson observations" in Proc. IEEE Conf. Decision and Control, Phoenix, AZ, Dec. 1999, pp. 4106-4110.
19. W. P. Malcolm, R. J. Elliott, and J. van der Hoek, "On the numerical stability of time-discretised state estimation via clark transformations" in Proc. IEEE Conf. Decision and Control, Maui, HI, Dec. 2003, pp. 1406-1412.
20. C. Moler and C. V. Loan, "Nineteen dubious ways to compute the matrix exponential, twenty-five years later" SIAM Rev., vol. 45, no. 1, pp. 1-39, 2003.
21. J. Moyal, "Stochastic processes and statistical physics" in Proc. Royal Statistical Soc., 1949, pp. 150-210.
22. E. Pardoux, "Equations du filtrage nonlineare de la predictions et du lissage" Stochastics, no. 6, pp. 193-231, 1982.
23. E. Pardoux and P. Protter, "A two sided stochastic integral and its calculus" Probab. Theory Related Fields, no. 76, pp. 15-49, 1987.
24. Filtering of a Diffusion Process With Poisson Type Observation, ser. Lecture Notes in Control and Information Sciences, pp. 510-518, 1977.
25. M. H. A. Davis, A. Segall, and T. Kallath, "Nonlinear filtering with counting observations" IEEE Trans. Inf. Theory, vol. 21, no. 2, pp. 143-149, Mar. 1975.
26. D. L. Snyder, "Filtering and detection for doubly stochastic poisson processes" IEEE Trans. Inf. Theory, vol. IT-18, no. 1, pp. 91-102, Jan. 1972.
27. D. L. Snyder, "Smoothing for doubly stochastic poisson processes" IEEE Trans. Inf. Theory, vol. AC-18, no. 5, pp. 558-562, Sep. 1972.
28. J. H. van Schuppen, "Filtering prediction and smoothing for counting process observations, a martingale approach" SIAM J. Appl. Math., vol. 32, no. 2, pp. 552-570, 1977.
29. D. R. Shin and E. I. Verriest, "The zakai-type formulas for counting process observations" presented at the Conf. Information Sciences and Systems, Princeton, NJ, 1990.
30. E. Wong and B. Hajek, Stochastic Processes in Engineering Systems. New York: Springer-Verlag, 1985.
31. Y. Zeng, "A partially observed model for micromovement of asset prices with bayes estimation via filtering" Math. Finance, vol. 13, no. 3, pp. 411-444, Jul. 2003.