Nonlinear filtering revisited: A spectral approach

SIAM Journal on Control and Optimization

Volumn 35, Issue 2, 1997, Pages 435-461

  Lototsky, Sergey ^a   Mikulevicius, Remigijus ^b   Rozovskii, Boris L ^a  

^a UNIVERSITY OF SOUTHERN CALIFORNIA   (United States)

^b VILNIUS UNIVERSITY   (Lithuania)

Author keywords

Cameron Martin development; Wick polynomials; Wiener chaos; Zakai equation

Indexed keywords

ALGORITHMS; COMPUTATIONAL METHODS; DIFFERENTIAL EQUATIONS; MARKOV PROCESSES; MATHEMATICAL OPERATORS; NUMERICAL METHODS; OPTIMAL CONTROL SYSTEMS; POLYNOMIALS; SIGNAL FILTERING AND PREDICTION; STOCHASTIC CONTROL SYSTEMS;

CAMERON-MARTIN METHOD; NONLINEAR FILTERING; WICK POLYNOMIALS; WIENER CHAOS; ZAKAI EQUATION;

CONTROL THEORY;

EID: 0031097261     PISSN: 03630129     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0363012993248918     Document Type: Article

References (38)

1. O. AXELSSON AND V. A. BARKER, Finite Element Solution of Boundary Value Problems, Academic Press, New York, 1984.
2. J. BARAS, Real-time architectures for the Zakai equations and applications, in Stochastic Analysis, E. Mayer-Wolf et al., eds., Academic Press, Boston, 1991.
3. V. E. BENESH, Exact finite-dimensional filters for certain diffusions with nonlinear drift, Stochastics, 5 (1981), pp. 65-92.
4. A. BENSOUSSAN, R. GLOWINSKI, AND R. RASCANU, Approximations of the Zakai equation by splitting up method, SIAM J. Control Optim., 28 (1990), pp. 1420-1431.
5. A. BUDHIRAJA AND G. KALLIANPUR, Approximations to the Solution of the Zakai Equations Using Multiple Wiener and Stratonovich Integral Expansions, Tech. report 447, Center for Stochastic Processes, University of North Carolina, Chapel Hill, NC, Jan. 1995.
6. Z. CAI AND WEINAN E, Hierarchical method for elliptic problems using wavelet, Comm. Appl. Numer. Methods, 8 (1992), pp. 819-825.
7. R. H. CAMERON AND W. T. MARTIN, The orthogonal development of non-linear functionals in a series of Fourier-Hermite functions, Ann. Math., 48 (1947), pp. 385-392.
8. J. M. C. CLARK, The design of robust approximations to the stochastic differential equations of nonlinear filtering, in Communication Systems and Random Processes Theory, NATO Adv. Sci. Inst. Ser. E Appl. Sci. 25, J. K. Skwirzynski, ed., Springer-Verlag, Berlin, 1977, pp. 721-734.
9. G. B. DIMASI AND W. J. RUNGGALDIER, On measure transformations for combined filtering and parameter estimation in discrete time, Systems Control Lett., 2 (1982), pp. 57-62.
10. R. J. ELLIOTT AND R. GLOWINSKI, Approximations to solutions of the Zakai filtering equation, Stochastic Anal. Appl., 7 (1989), pp. 145-168.
11. W. H. FLEMING AND E. PARDOUX, Optimal control for partially observed diffusions, SIAM J. Control Optim., 20 (1982), pp. 261-285.
12. P. FLORCHINGER AND F. LEGLAND, Time discretization of the Zakai equation for diffusion processes observed in correlated noise, Stochastics Stochastics Rep., 35 (1991), pp. 233-256.
13. C. P. FUNG, Solving Hidden Markov Problems by Spectral Methods, Ph.D. thesis, University of Southern California, Los Angeles, CA, Nov. 1995.
14. J. GLIMM AND A. JAFFE, Quantum Physics, Springer, New York, 1987.
15. D. GOTTLIEB AND S. A. ORSZAG, Numerical Analysis of Spectral Methods: Theory and Applications, CBMS-NSF Regional Conf., Ser. in Appl. Math., 26, SIAM, Philadelphia, 1977.
16. T. HIDA, H.-H. KUO, J. POTTHOFF, AND L. SREIT, White Noise, Kluwer, Boston, 1993.
17. K. ITO, Multiple Wiener integral, J. Math. Soc. Japan, 3 (1951), pp. 157-169.
18. K. ITO, Approximation of the Zakai equation for nonlinear filtering, SIAM J. Control Optim., 34 (1996), pp. 620-634.
19. G. KALLIANPUR, Stochastic Filtering Theory, Springer-Verlag, New York, 1980.
20. N. V. KRYLOV AND B. L. ROZOVSKII, On conditional distributions of diffusion processes, Math. USSR-Izv., 42 (1978), pp. 336-356.
21. N. V. KRYLOV AND A. VERETENNIKOV, On explicit formulae for solutions of stochastic equations, Math. USSR-Sb., 29 (1976), pp. 239-256.
22. H. KUNITA, Cauchy problem for stochastic partial differential equations arising in non-linear filtering theory, Systems Control Lett., 1 (1981), pp. 37-41.
23. H. KUNITA, Stochastic Flows and Stochastic Differential Equations, Cambridge University Press, Cambridge, UK, 1982.
24. H. J. KUSHNER, Probability Methods for Approximations in Stochastic Control and for Elliptic Equations, Academic Press, New York, 1977.
25. O. A. LADYZHENSKAIA, V. A. SOLONIKOV, AND N. N. URAL'TSEVA, Linear and Quasi-linear Equations of Parabolic Type, American Mathematical Society, Providence, RI, 1968.
26. F. LEGLAND, Time discretizations of nonlinear filtering equations, in Proc. 28th IEEE Conf. on Decision and Control, Tampa, 1989, IEEE Control Systems Society, Piscataway, NJ, 1989, pp. 2601-2606.
27. R. S. LIPTSER AND A. N. SHIRYAYEV, Statistics of Random Processes, Springer-Verlag, New York, 1992.
28. J. T.-H. LO AND S.-K. NG, Optimal orthogonal expansion for estimation I: Signal in white Gaussian noise, in Nonlinear Stochastic Problems, R. Bucy and J. Moura, eds., D. Reidel, Dordrecht, 1983, pp. 291-309.
29. R. MIKULEVICIUS AND B. L. ROZOVSKII, Linear parabolic stochastic PDEs and Wiener chaos, unpublished.
30. R. MIKULEVICIUS AND B. L. ROZOVSKII, Separation of observations and parameters in non-linear filtering, in Proc. 32nd IEEE Conf. on Decision and Control, Part 2, San Antonio, 1993, IEEE Control Systems Society, Piscataway, NJ, 1993, pp. 1564-1569.
31. R. MIKULEVICIUS AND B. L. ROZOVSKII, Soft solutions of linear parabolic SPDE's and the Wiener chaos expansion, in Stochastic Analysis on Infinite Dimensional Spaces, Pitman Res. Notes Math. Sec. 310, H. Kunita and H.-H. Kuo, eds., Longman, Harlow, UK, 1994.
32. D. OCONE, Multiple integral expansions for nonlinear filtering, Stochastics, 10 (1983), pp. 1-30.
33. E. PARDOUX, Filtrage non lineaire et equations aux derivees partielles stochastiques associees, in Ecole d'été de Probabilités de Saint-Flour, Springer-Verlag, New York, 1989.
34. B. L. ROZOVSKII, Stochastic Evolution Systems, Kluwer, Amsterdam, 1990.
35. E. WONG, Explicit solutions to a class of nonlinear filtering problems, Stochastics, 5 (1981), pp. 311-321.
36. D. M. YOUNG, Iterative Solution of Large Linear Systems, Academic Press, New York, 1971.
37. M. ZAKAI, On the optimal filtering of diffusion processes, Z. Wahrscheinlichkeitstheorie und verw. Gebiete, 4 (1969), pp. 230-233.
38. A. K. ZVONKIN AND N. V. KRYLOV, Strong solutions of stochastic differential equations, in Proc. of the School Seminar on the Theory of Random Processes, Druskinikai, November 25-30, 1974, Vilnius, 1975 (in Russian).