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Exact rates of convergence for a branching particle approximation to the solution of the Zakai equation. (English) Zbl 1137.60335

Summary: In [D. Crisan, J. Gaines and T. J. Lyons, SIAM J. Appl. Math. 58, No. 5, 1568–1590 (1998; Zbl 0915.93060)] we describe a branching particle algorithm that produces a particle approximation to the solution of the Zakai equation and find an upper bound for the rate of convergence of the mean square error. In this paper, the exact rate of convergence of the mean square error is deduced. Also, several variations of the branching algorithm with better rates of convergence are introduced.

MSC:

60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
60G07 General theory of stochastic processes
60G35 Signal detection and filtering (aspects of stochastic processes)
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60H30 Applications of stochastic analysis (to PDEs, etc.)
93E11 Filtering in stochastic control theory

Citations:

Zbl 0915.93060
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References:

[1] BENSOUSSAN, A. (1992). Stochastic Control of Partially Observable Sy stems. Cambridge Univ. Press. · Zbl 0776.93094
[2] CRISAN, D. (1996). The problem of nonlinear filtering. Ph.D. thesis, Edinburgh.
[3] CRISAN, D., DEL MORAL, P. and Ly ONS, T. (1999). Discrete filtering using branching and interacting particle sy stems. Markov Process. Related Fields 5 293-318. · Zbl 0967.93088
[4] CRISAN, D., GAINES, J. and Ly ONS, T. (1998). Convergence of a branching particle method to the solution of the Zakai equation. SIAM J. Appl. Probab. 58 1568-1591. JSTOR: · Zbl 0915.93060
[5] CRISAN, D. and Ly ONS, T. (1997). Nonlinear filtering and measure valued processes. Probab. Theory Related Fields 109 217-244. · Zbl 0888.93056
[6] CRISAN, D. and Ly ONS, T. (1999). A particle approximation of the solution of the Kushner- Stratonovitch equation. Probab. Theory Related Fields 115 549-578. · Zbl 0951.93068
[7] DOUCET, A., DE FREITAS, N. and GORDON, N. (2001). Sequential Monte Carlo Methods in Practice. Statistics for Engineering and Information Science. Springer, New York. · Zbl 0967.00022
[8] ETHIER, S. N. and KURTZ, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York. · Zbl 0592.60049
[9] FUJISAKI, M., KALLIANPUR, G. and KUNITA, H. (1972). Stochastic differential equations for the non linear filtering problem. Osaka J. Math. 9 19-40. · Zbl 0242.93051
[10] KALLIANPUR, G. and STRIEBEL, C. (1968). Estimation of stochastic sy stems: Arbitrary sy stem process with additive white noise observation errors. Ann. Math. Statist. 39 785-801. · Zbl 0174.22102
[11] KARATZAS, I. and SHREVE, S. E. (1988). Brownian Motion and Stochastic Calculus. Springer, New York. · Zbl 0638.60065
[12] KRy LOV, N. V. and ROZOVSKII, B. L. (1978). Conditional distributions of diffusion processes. Izv. Akad. Nauk SSSR Ser. Mat. 42 356-378. · Zbl 0402.60077
[13] KURTZ, T. G. and OCONE, D. L. (1988). Unique characterisation of conditional distributions in nonlinear filtering. Ann. Probab. 16 80-107. · Zbl 0655.60035
[14] KURTZ, T. G. and XIONG, J. (1999). Particle representation for a class of nonlinear SPDE’s. Stochastic Process. Appl. 83 103-106. · Zbl 0996.60071
[15] KURTZ, T. G. and XIONG, J. (2001). Numerical solutions for a class of SPDEs with application to filtering. In Stochastics in Finite and Infinite Dimensions (T. Hida, R. L. Karandikar, H. Kunita, B. S. Rajput, S. Watanabe and J. Xiong, eds.) 233-258. Birkhäuser, Boston. · Zbl 0991.60053
[16] KUSHNER, H. J. (1967). Dy namical equations for optimal nonlinear filtering. J. Differential Equations 3 179-190. · Zbl 0158.16801
[17] PARDOUX, E. (1979). Stochastic partial differential equations and filtering of diffusion processes. Stochastics 3 127-167. · Zbl 0424.60067
[18] PARDOUX, E. (1991). Filtrage non linéaire et equations aux dérivées partielles stochastiques associées. Ecole d’Eté de Probabilités de Saint-Flour XIX. Lecture Notes in Math. 1464. Springer, New York. · Zbl 0732.60050
[19] YOR, M. (1977). Sur les théories du filtrage et de la prédiction. Séminaire de Probabilités XI. Lecture Notes in Math. 581 257-297. Springer, New York.
[20] ZAKAI, M. (1969). On the optimal filtering of diffusion processes. Z. Wahrsch. Verw. Gebiete 11 230-243. · Zbl 0164.19201
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