Summary
Let (ρ t ϕ)0≦t≦1 be the unnormalized filter arising in the filtering theory of correlated diffusions. In this article, ϱ. φ. is considered as a stochastic process taking values inC(ℝn,ℝ); a description of the support of its law in the Fréchet spaceC([0,1],C(ℝn,R)) is given. This result is the analogue for stochastic partial differential equations of the celebrated Stroock-Varadhan theorem for diffusion processes. The support of the law of the filter is shown to be the closure of the set of trajectories obtained from the Zakai equation by replacing the Stratonovitch differentialdy by anH 1-control (herey denotes the observation process).
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Ball, J.M., Marsden, J.E., Slemrod, M.: Controllability for distributed bilinear systems. SIAM J. Control Optimization20, 575–597 (1982)
Bernard, P.: Propriétés des trajectoires des fonctions aléatoires stables sur ℝd. Ann. I.H.P.,VI (2), 131–151 (1970)
Bismut, J.M.: Martingales, the Malliavin calculus and hypoellipticity under general Hörmander's conditions. Z. Wahrscheinlichkeitstheorie Verw. Geb.56, 469–505 (1981)
Bismut, J.M.: Mécanique aléatoire. (Lect. Notes Math., vol. 866) Berlin Heidelberg New York: Springer (1981)
Bony, J.M.: Principe du maximum, inégalité de Harnack et unicité du probléme de Cauchy pour les opérateurs elliptiques dégénérés. Ann. Inst. Fourier19, 277–304 (1969)
Brezis, H.: Analyse fonctionnelle, Paris: Masson 1983
Brocket, R.W.: Non linear systems and non linear estimation theory. In: Hazewinkel, Willems (eds.) Stochastic Systems: the mathematics of filtering and identification and applications. Dordrecht: Reidel 1981
Chaleyat-Maurel, M., Michel, D.: Une propriété de robustesse en filtrage non linéaire. Stochastics19, 11–40 (1986)
Chaleyat-Maurel, M., Michel, D.: The support of the law of a filter inC ∞ topology. Proceedings of IMA Workshop of Stochastic Diff. Systems, Stochastic Control Theory and Application. (Minneapolis)10, 395–409 (1988)
Chaleyat-Maurel, M., Michel, D.: The support of the density of the filter in the uncorrelated case. Proceedings of the 1988 Trento Conference on Stochastic Partial Differential Equations. G. Da Prato, L. Tubaro (eds.). To appear
Di Masi, G.B., Runggaldier, W.J., Armellin, B.: On recursive approximations with error bounds in non linear filtering. International conference on stochastic optimization. Kiev. USSR (1984)
Di Masi, G.B., Pratelli, M., Runggaldier, W.J.: An approximation for the non linear filtering problem with error bound. Stochastics14, 247–272 (1985)
Ikeda, N., Watanabe, S.: Stochastic differential equations and diffusion processes. Amsterdam-Tokyo: North Holland/Kodansha 1981
Korezlioglu, H., Mazziotto, G.: Modelization and filtering of discrete systems and discrete approximation of continuous systems. “System modelling and Optimization”. (Lect. Notes Control Inf. Sci., vol. 59) Berlin Heidelberg New York: Springer 1984
Korezlioglu, H., Mazziotto, G.: Approximation of the non linear filter by periodic sampling and quantization. “Analysis and Optimization of systems”. (Lect. Notes Control Inf. Sci., vol. 63) Berlin Heidelberg New York: Springer 1984
Kunita, H.: Stochastic differential equations and stochastic flows of diffeomorphisms. Ecole d'été de Probabilités de St Flour XII 1982 (Lect. Notes Math., vol. 1097, pp 114–305) Berlin Heidelberg New York: Springer 1984
Kunita, H.: Convergence of stochastic flows connected with stochastic Ordinary Differential Equations. Stochastics17, 215–251 (1986)
Kushner, H.J.: Probability methods for approximations in stochastic control and elliptic equations. New York: Academic Press 1977
Kushner, H.H., Huang, H.: Approximate and limit results for non linear filters with wide bandwidth observation noise. Stochastics16, 65–96 (1986)
Le Gland, F.: Estimation de paramètres dans les processus stochastiques, en observation incomplète. Application à un problème de Radio-Astronomie. Thèse de Docteur-Ingénieur, Université Paris 9 (1981)
Malliavin, P.: Stochastic calculus of variations and hypoelliptic operators. In: Ito, K. (ed.) Proc. of the International Symp. on stochastic differential equations. New York: Wiley 1978
Marcus, S.I.: Modelling and approximation of stochastic differential equations driven by semi-martingales. Stochastics4, 223–245 (1981)
Moulinier, J.M.: Théorèmes limites pour les processus à un ou deux paramètres. “Thèse de 3ème cycle. Université de Paris VI (1982)
Mouliner, J.M.: Théorèmes limites pour les équations différentielles stochastiques. Bull. Sci. Math., II. Ser.112, 118–210 (1988)
Ocone, D.: Application of Wiener space analysis to non linear filtering. Proc. of the Int. Symp. on Math. Systems and Networks. Stockholm (1985)
Ocone, D.: Probability distributions of solutions to some stochastic partial differential equations. In: Da Prato, G., Tubaro L. (eds.) Stochastic Partial Differential Equations and Applications. (Lect. Notes Math., vol. 1236) Berlin Heidelberg New York: Springer 1987
Ocone, D.: Stochastic calculus of variations for stochastic partial differential equations. J.F.A.79, 288–331 (1988)
Picard, J.: Approximations of non linear filtering problems and order of convergence. Filtering and control of random processes. (Lect. Notes Control Inf. Sci., vol. 61) Berlin Heidelberg New York: Springer 1984
Picard, J.: An estimate of the error in time discretization of non linear filtering problems. Proc. of the Int. Symp. on math. systems and Networks. Stockholm (1985)
Picard, J.: Convergence in probability for perturbed stochastic integral equations. Probab. Th. Rel. Fields81, 383–451 (1989)
Priouret, P.: Diffusions et équations différentielles stochastiques. Ecole d'été de Saint-Flour III 1973. (Lect. Notes Math., vol. 390) Berlin Heidelberg New York: Springer 1974
Sznitman, A.S.: Martingales dépendant d'un paramètre. Une formule d'Itô. Z. Wahrscheinlichkeitstheor. Verw. Geb.60, 41–70 (1982)
Stroock, D.W., Varadhan, S.: On the support of diffusion processes with applications to the strong maximum principle. Proc. 6th Berkeley Symp. Math. Stat. Probabilités III, pp. 333–359. Berkeley: University California Press 1972
Wong, E., Zakai, M.: On the convergence of ordinary integrals to stochastic integrals. Ann. Math. Stat.36, 1560–1564 (1965)
Zakai, M.: On the optimal filtering of diffusion processes. Z. Zahrscheinlichkeitstheor. Verw. Geb.11, 230–243 (1969)
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Chaleyat-Maurel, M., Michel, D. A Stroock Varadhan support theorem in non-linear filtering theory. Probab. Th. Rel. Fields 84, 119–139 (1990). https://doi.org/10.1007/BF01288562
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DOI: https://doi.org/10.1007/BF01288562