Stochastic calculus of variations for stochastic partial differential equations. (English) Zbl 0653.60046
Malliavin calculus is developed for some abstract evolution equations which satisfy an appropriate coercivity condition. Since these equations generally contain unbounded operators in Hilbert spaces, Galerkin approximations are used for the proof of the smoothness of correlation functionals on Wiener spaces [cf. E. Pardoux, Stochastics 3, 127- 167 (1979; Zbl 0424.60067)].
The above results are applied to a class of stochastic pde’s which includes the Zakai equation of nonlinear filtering. The main theorem of this part of the paper states a Lie algebraic criterion for the existence of a density for the finite-dimensional projections of the solution of the stochastic pde. The proof is related to the known proofs of Hörmander’s hypoellipticity theorem using Malliavin’s calculus.
The above results are applied to a class of stochastic pde’s which includes the Zakai equation of nonlinear filtering. The main theorem of this part of the paper states a Lie algebraic criterion for the existence of a density for the finite-dimensional projections of the solution of the stochastic pde. The proof is related to the known proofs of Hörmander’s hypoellipticity theorem using Malliavin’s calculus.
Reviewer: P.Kröger
MSC:
| 60H07 | Stochastic calculus of variations and the Malliavin calculus |
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
| 60G35 | Signal detection and filtering (aspects of stochastic processes) |
Keywords:
Malliavin calculus; coercivity condition; Galerkin approximations; Zakai equation of nonlinear filtering; Hörmander’s hypoellipticity theoremCitations:
Zbl 0424.60067References:
| [1] | Ball, J. M.; Marsden, J. E.; Slemrod, M., Controllability for distributed bilinear systems, SIAM J. Control Optim., 20, 575-597 (1982) · Zbl 0485.93015 |
| [2] | Bardos, C.; Tartar, L., Sur l’unicite des equation paraboliques et quelques questions voisines, Arch. Rational Mech. Anal., 50, 10-25 (1973) · Zbl 0258.35039 |
| [3] | Bismut, J. M., Martingales, the Malliavin calculus and hypoellipticity under general Hormander’s conditions, Z. Wahrsch. Verw. Gebiete, 56, 469-505 (1981) · Zbl 0445.60049 |
| [4] | Bensoussan, A.; Lions, J. L., Applications of Variational Inequalities in Stochastic Control (1982), North-Holland: North-Holland Amsterdam · Zbl 0478.49002 |
| [5] | Bouc, R.; Pardoux, E., Moments of semilinear random evolutions, SIAM J. Appl. Math., 41, 370-399 (1981) · Zbl 0469.60060 |
| [6] | Chaleyat-Maurel, M., Robustesse en theorie du filtrage non lineaire et calcul des variations stochastique, C. R. Acad. Sci. Paris, 297, 541-544 (1983) · Zbl 0535.60033 |
| [7] | Ferreyra, G. S., Smoothness of the unnormalized conditional measures of stochastic nonlinear filtering, (Proceedings of the 23rd IEEE Conference on Decision and Control. Proceedings of the 23rd IEEE Conference on Decision and Control, Las Vegas (1984)), 705-711 |
| [8] | Friedman, A., Partial Differential Equations of Parabolic Type (1964), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ · Zbl 0144.34903 |
| [9] | Krylov, N.; Rozovskii, On the conditional distribution of diffusion processes, Math. USSR-Izv., 42 (1978) · Zbl 0402.60077 |
| [10] | Kusuoka, S.; Stroock, D. M., The partial Malliavin calculus and its application to nonlinear filtering, Stochastics, 12, 83-142 (1984) · Zbl 0567.60046 |
| [11] | Kusuoka, S.; Stroock, D. W., Applications of the Malliavin calculus, (Ito, K., Proceedings, 1982 Taniguchi Conference in Katata. Proceedings, 1982 Taniguchi Conference in Katata, Japan (1982), Kinikuniya: Kinikuniya Tokyo) · Zbl 0568.60059 |
| [12] | Marcus, S. I., Algebraic and geometric methods in nonlinear filtering, SIAM J. Control Optim., 22, 817-844 (1984) · Zbl 0548.93073 |
| [13] | Malliavin, P., Stochastic calculus of variations and hypoelliptic operators, (Ito, K., Proceedings, International Symposium on S.D.E.. Proceedings, International Symposium on S.D.E., Kyoto (1978), Kinokuniya: Kinokuniya Tokyo) · Zbl 0411.60060 |
| [14] | Malliavin, P., \(C^k\)-hypoellipticity with degeneracy, (Stochastic Analysis (1978), Academic Press: Academic Press New York/London), 327-340 · Zbl 0449.58022 |
| [15] | Michel, D., Régularité des lois conditionelles en théorie du filtrage non-linéaire et calcul des variations stochastique, J. Funct. Anal., 41, 8-36 (1981) · Zbl 0487.60053 |
| [16] | Nualart, D.; Zakai, M., Generalized stochastic integrals and the Malliavin calculus, Probability Theory and Related Fields, 73, 255-280 (1986) · Zbl 0601.60053 |
| [17] | Pardoux, E., Stochastic partial differential equations and filtering of diffusion processes, Stochastics, 3, 127-167 (1979) · Zbl 0424.60067 |
| [18] | Shigekawa, I., Derivatives of Wiener functionals and absolute continuity of induced measures, J. Math. Kyoto Univ., 20, 263-289 (1980) · Zbl 0476.28008 |
| [19] | Stroock, D. W., The Malliavin calculus, a functional analytic approach, J. Funct. Anal., 44, 212-257 (1981) · Zbl 0475.60060 |
| [20] | Stroock, D. W., Some applications of stochastic calculus to partial differential equations, (Ecole d’Eté de Probabilités de Saint-Fleur XI, 1981. Ecole d’Eté de Probabilités de Saint-Fleur XI, 1981, Lecture Notes in Mathematics, Vol. 976 (1983), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0552.60022 |
| [21] | Watanabe, S., Stochastic Differential Equations and Malliavin Calculus (1984), Tata Institute of Fundamental Research, Springer-Verlag: Tata Institute of Fundamental Research, Springer-Verlag New York/Berlin · Zbl 0546.60054 |
| [22] | Zakai, M., The Malliavin calculus, Acta Appl. Math., 3, 175-207 (1985) · Zbl 0553.60053 |
| [23] | Bismut, J. M.; Michel, R., Diffusions conditionelles, J. Funct. Anal., 45, 274-282 (1982) · Zbl 0475.60062 |
| [24] | Meyer, P. A., Quelques résultats analytiques sur le semi-groupe d’Ornstein-Uhlenbeck en dimension infinie, (Kallianpur, G., Theory and Applications of Random Fields. Theory and Applications of Random Fields, Lecture Notes in Control and Information Sciences, Vol. 49 (1983), Springer-Verlag: Springer-Verlag New York/Berlin) · Zbl 0514.60076 |
| [25] | Föllmer, H., Calcul d’Ito sans probabilité, (Seminaire de Probabilités XV. Seminaire de Probabilités XV, Lecture Notes in Mathematics, Vol. 850 (1980), Springer-Verlag: Springer-Verlag New York/Berlin), 143-150 · Zbl 0461.60074 |
| [26] | Bouleau, N.; Hirsch, F., Calculs functionnels dans les espaces de Dirichlet et application aux équations differentielles lipschitziennes, C. R. Acad. Sci. Paris Sér. I Math., 302, 263-266 (1986) · Zbl 0604.60072 |
| [27] | Moulinier, J. M., Absolue continuité de probabilités de transition par rapport à une mesure gaussienne dans un espace de Hilbert, J. Funct. Anal., 275-295 (1985) · Zbl 0624.60055 |
| [28] | Ocone, D., Probability Densities for Conditional Statistics in the Cubic Sensor Problem, Mathematics of Control, Signals, and Systems, 1 (1988) · Zbl 0658.60085 |
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