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Abstrakty
In this paper, we prove, using Malliavin calculus, that under a global Hörmander condition the law of a Riemannian manifold valued stochastic process, a solution of a stochastic differential equation with time dependent coefficients, admits a C∞-density with respect to the Riemannian volume element. This result is applied to a nonlinear filtering problem with time dependent coefficients on manifolds.
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Czasopismo
Rocznik
Tom
Strony
319--334
Opis fizyczny
Bibliogr. 19 poz.
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autor
- U.R.A. C.N.R.S. No 399, Departement de Mathématiques, Université de Metz, BP 80794, F-57012 Metz Cedex, France
Bibliografia
- [1] D. Вakry, Un critere de non-explosion pour certaines diffusions sur une variété riemannienne complete, C. R. Acad. Sei. Sér. I, 303 (1986), pp. 23-26.
- [2] J. M. Bismut, Martingales, the Malliavin calculus and hypoellipticity under general conditions, Z. Wahrsch. Verw. Gebiete 56 (1981), pp. 469-505.
- [3] M. Chaleyat-Maurel and D. Michel, Hypoellipticity theorems and conditional laws, ibidem 65 (1984), pp. 573-597.
- [4] Т. E. Duncan, Some filtering results in Riemannian manifolds, Inform, and Control 35 (1977), pp. 182-195.
- [5] K. D. El worthy, Stochastic Differential Equations on Manifolds, London Math. Soc. Lecture Note Ser. 70, Cambridge University Press, 1984.
- [6] P. Florchinger, Malliavin calculus with time depending coefficients and application to nonlinear filtering, Probab. Theory Related Fields 86 (1990), pp. 203-233.
- [7] — Existence of a smooth density for the filter in nonlinear filtering on manifolds, in: Partial Differential Equations and Their Applications, B. Rozovskii and R. Sowers (eds.), Lecture Notes in Control and Inform. Sei. 176, Springer, 1992.
- [8] N Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, 2nd edition, North-Holland-Kadansha, 1989.
- [9] S. Kusuoka and D. W. Stroock, Applications of the Malliavin Calculus. Part T. Taniguchi Symp. (Katata-Kyoto 1982), K. Ito (ed.), North-Holland, Amsterdam-Oxford-New York1984, pp. 277-306. Part II: J. Fac. Sci. Univ. Tokyo Sect. IA 32 (1985), pp. 1-76.
- [10] X. M. Li, Properties at infinity of diffusion semigroups and stochastic flows via weak uniform covers, J. Potential Analysis 3 (1994), pp. 339-357.
- [11] P. Malliavin, Stochastic calculus of variations and hypoelliptic operators, in: Proceedings of the International Conference of Stochastic Differential Equations 1976, Kyoto, Kinokuniya-Wiley, Tokyo-New York 1978, pp. 195-263.
- [12] S. K. Ng and P. E. Caines, Nonlinear filtering in Riemannian manifolds, IMA J. Control. Inform. 2 (1985), pp. 25-36.
- [13] J. Norris, Simplified Malliavin calculus, in: Séminaire de Probabilités XX, J. Azéma and M. Yor (Eds.), Lecture Notes in Math. 1204, Springer, Berlin-Heidelberg-New York 1986, pp. 101-130.
- [14] D. Nualart, The Malliavin Calculus and Related Topics. Probabilities and Their Applications, Springer, New York-Berlin-Heidelberg 1995.
- [15] M. Pontier and J. Spzirglas, Filtering with observation on a Riemannian symmetric space, SIAM J. Control Optim. 26 (3) (1988), pp. 609-627.
- [16] D. Stroock, Some applications of stochastic calculus to partial differential equations, in: Ecole d’Eté de Probabilités de Saint Flour, P. L. Hennequin (Ed.), Lecture Notes in Math. 976, Springer, Berlin-Heidelberg-New York 1983, pp. 267-382.
- [17] S. Tanigushi, Malliavin’s stochastic calculus of variations for manifold-valued Wiener functionals and its applications, Z. Wahrsch. Verw. Gebiete 65 (1983), pp. 269-290.
- [18] M. Zakai, On the optimal filtering of diffusion processes, ibidem 11 (1969), pp. 230-243.
- [19] — The Malliavin calculus, Acta Appl. Math. 3-2 (1985), pp. 175-207.
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Bibliografia
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