Observation sampling and quantisation for continuous-time estimators

Stochastic Processes and their Applications

Volumn 87, Issue 2, 2000, Pages 311-337

  Newton, Nigel J ^a  

^a UNIVERSITY OF ESSEX   (United Kingdom)

Author keywords

60H10; 65C05; 65M99; 93E20; Approximation; Continuous time Bayesian estimation; Nonlinear filtering; Over sampling; Quantisation; Weak convergence

Indexed keywords

EID: 0041707111     PISSN: 03044149     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0304-4149(99)00117-9     Document Type: Article

References (33)

1. Bennaton J.F. Discrete-time Galerkin approximations to the nonlinear filtering solution. J. Math. Anal. Appl. 110:1985;364-383.
2. Bucy R.C. Nonlinear filtering theory. IEEE Trans. Automat. Control. 10:1965;198-199.
3. Budhiraja, A., Kushner, H.J., 1998. Approximation and limit results for nonlinear filters over an infinite time horizon. Preprint, Brown University, Providence.
4. Chaleyat-Maurel M. Robustesse du filtre et calcul des variations stochastiques. J. Funct. Anal. 68:1986;55-71.
5. Chaleyat-Maurel M., Michel D. Une propriété de continuité en filtrage nonlinéaire. Stochastics. 19:1986;11-40.
6. Clark, J.M.C., 1978. The design of robust approximations to the stochastic differential equations of nonlinear filtering. In: J.K. Skwirzynski (Ed.), Communication Systems and Random Process Theory, NATO Advanced Study Institute Series. Sijthoff and Noordhoff, Alphen aan den Rijn, pp. 721-734.
7. Crisan D., Lyons T. Nonlinear filtering and measure-valued processes. Probab. Theory Related Fields. 109:1997;217-244.
8. Crisan, D., Lyons, T., 1998. A particle approximation of the solution of the Kushner-Stratonovich equation. Preprint, Imperial College, London.
9. Davis M.H.A. A pathwise solution to the equations of nonlinear filtering. Theory Probab. Appl. 27:1982;167-174.
10. Dellacherie C., Meyer P.A. Probabilités et Potentiel, Chapitres V à VIII: Théorie des Martingales. 1980;Hermann, Paris.
11. Di Masi G.B., Pratelli M., Runggaldier W.J. An approximation for the nonlinear filtering problem with error bound. Stochastics. 14:1985;247-271.
12. Elliott R.J., Kohlmann M. Robust filtering for correlated multidimensional observations. Math. Z. 178:1981;559-578.
13. Ethier S.N., Kurtz T.G. Markov Processes: Characterisation and Convergence. 1986;Wiley, New York.
14. Fleming W.H., Mitter S.K. Optimal control and nonlinear filtering for nondegenerate diffusion processes. Stochastics. 8:1982;63-77.
15. Fujisaki M., Kallianpur G., Kunita H. Stochastic differential equations for the nonlinear filtering problem. Osaka J. Math. 9:1972;19-40.
16. Kallianpur G., Striebel C. Estimation of stochastic systems: arbitrary system process with additive white noise observation errors. Ann. Math. Statist. 39:1968;785-801.
17. Korezlioglu H., Mazziotto G. Approximations of the nonlinear filter by periodic sampling and quantisation. Bensoussan A., Lions J.L. Analysis and Optimisation of Systems, Part 1. 1984;553-567 Springer, Berlin.
18. Kushner H.J. On the differential equations satisfied by conditional probability densities of Markov processes. SIAM J. Control. 2:1964;106-119.
19. Kushner H.J. Dynamical equations for nonlinear filtering. J. Differential Equations. 3:1967;179-190.
20. Kushner H.J. A robust discrete-state approximation to the optimal nonlinear filter for a diffusion. Stochastics. 3:1979;75-83.
21. Le Gland, F., 1992. Splitting-up approximation for SPDEs and SDEs with application to nonlinear filtering. Lectures Notes in Control and Information Sciences, Vol. 176, pp. 177-187.
22. Le Gland F. Refined and high order time discretization of nonlinear filtering equations. Proceedings of the 28th IEEE CDC. pp. 2601-2606:1989.
23. Liptser R.S., Shiryayev A.N. Statistics of Random Processes 1 - General Theory. 1977;Springer, New York.
24. Milshtein G.N. Approximate integration of stochastic differential equations. Theory Probab. Appl. 19:1974;557-562.
25. Newton N.J. An asymptotically efficient difference formula for solving stochastic differential equations. Stochastics. 19:1986;175-206.
26. Newton, N.J., 2000. Observations pre-processing and quantization for nonlinear filters. SIAM J. Control Optim. 38, 482-502.
27. Picard J. Approximation of nonlinear filtering problems and order of convergence. Korezlioglu H., Mazziotto G., Szpirglas J. Filtering and Control of Random Processes (E.N.S.T.-C.N.E.T. 1983). 1984;219-236 Springer, Berlin.
28. Picard J. Robustesse de la solution des problemes de filtrage avec bruit blanc independent. Stochastics. 13:1984;229-245.
29. Stratonovich R.L. Conditional Markov processes. Theory Probab. Appl. 5:1960;156-178.
30. Stricker C., Yor M. Calcul stochastique dépendant d'un parametre. Z. Wahrsch. Verw. Geb. 45:1978;109-133.
31. Sussmann H. Continuous versions of the conditional statistics of nonlinear filtering. Stochastic Differential Systems: Filtering and Control, Proceedings of the IFIP-WG. 7/1 Working Conference, Marseille-Luminy, Lecture Notes in Control and Information Sciences Vol. 69:1985;pp 235-246.
32. Wonham W.M. Some applications of stochastic differential equations to optimal nonlinear filtering. SIAM J. Control. 2:1965;347-369.
33. Zakai M. On the optimal filtering of diffusion processes. Z. Wahrsch. Geb. 11:1969;230-243.