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On uniqueness of solutions for the stochastic differential equations of nonlinear filtering. (English) Zbl 1017.60048

The authors study the uniqueness for the stochastic differential equations of nonlinear filtering from a point of view similar to that of J. Szpirglas [Ann. Inst. Henri Poincaré, n. Sér., Sect. B 14, 33-59 (1978; Zbl 0374.60055)], but for a nonlinear filtering problem in which there is dependence of the signal on the observations.

MSC:

60G35 Signal detection and filtering (aspects of stochastic processes)
60G44 Martingales with continuous parameter

Citations:

Zbl 0374.60055
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References:

[1] Bhatt, A. G., Kallianpur, G. and Karandikar, R. L. (1995). Uniqueness and robustness of solution of measure-valued equations of nonlinear filtering. Ann. Probab. 23 1895- 1938. · Zbl 0861.60051
[2] Bhatt, A. G. and Karandikar, R. L. (1995). Evolution equations for Markov processes: application to the white-noise theory of filtering. Appl. Math. Optim. 31 327-348. · Zbl 0824.60075
[3] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York. · Zbl 0592.60049
[4] Ikeda, N. and Watanabe, S. (1989). Stochastic Differential Equations and Diffusion Processes. North-Holland, Amsterdam. · Zbl 0684.60040
[5] Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus. Springer, New York. · Zbl 0734.60060
[6] Kunita, H. (1971). Asymptotic behavior of the nonlinear filtering errors of Markov processes. J. Multivariate Anal. 1 365-393. · Zbl 0245.93027
[7] Kunita, H. (1981). Stochastic partial differential equations connected with nonlinear filtering. Lecture Notes in Math. 972 100-169. Springer, Berlin. · Zbl 0527.60067
[8] Kunita, H. (1990). Stochastic Flows and Stochastic Differential Equations. Cambridge Univ. Press. · Zbl 0743.60052
[9] Kurtz, T. G. (1998). Martingale problems for conditional distributions of Markov processes. Electron J. Probab. 3 1-29. · Zbl 0907.60065
[10] Kurtz, T. G. and Ocone, D. L. (1988). Unique characterization of conditional distributions in nonlinear filtering. Ann. Probab. 16 80-107. · Zbl 0655.60035
[11] Kurtz, T. G. and Stockbridge, R. H. (1988). Existence of Markov controls and characterization of optimal Markov controls. SIAM J. Control Optim. 36 609-653. · Zbl 0935.93064
[12] Revuz, D. and Yor, M. (1994). Continuous Martingales and Brownian Motion. Springer, New York. · Zbl 0804.60001
[13] Rogers, L. C. G. and Williams, D. (1987). Diffusions, Markov Processes and Martingales 2: It o Calculus. Wiley, New York. · Zbl 0977.60005
[14] Rozovskii, B. L. (1991). A simple proof of uniqueness for Kushner and Zakai equations. In Stochastic Analysis (E. Mayer-Wolf, E. Merzbach and A. Schwartz, eds.) 449-458. Academic Press, New York. · Zbl 0732.60055
[15] Szpirglas, J. (1998). Sur l’équivalence d’équations différentielles stochastiques á valeurs mesures intervenant dans le filtrage markovien non linéaire. Ann. Inst. H. Poincaré Probab. Statist. 14 33-59. · Zbl 0374.60055
[16] Yamada, T. and Watanabe, S. (1971). On the uniqueness of solutions of stochastic differential equations. J. Math. Kyoto Univ. 11 155-167. · Zbl 0236.60037
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