Stochastic systems in Riemannian manifolds

  • T. E. Duncan
Contributed Papers

Abstract

A formulation of stochastic systems in a Riemannian manifold is given by stochastic differential equations in the tangent bundle of the manifold. Brownian motion is constructed in a compact Riemannian manifold as well as the horizontal lift of this process to the bundle of orthonormal frames. The solution of some stochastic differential equations in the tangent bundle of the manifold is defined by the transformation of the measure for the manifold-valued Brownian motion by a suitable Radon-Nikodym derivative. Real-valued stochastic integrals are defined for this Brownian motion using parallelism along the Brownian paths. A stochastic control problem is formulated and solved for these stochastic systems where a suitable convexity condition is assumed.

Key Words

Stochastic control problems stochastic games existence theorems sufficient conditions probability theory 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Benes, V. E.,Existence of Optimal Stochastic Control Laws, SIAM Journal on Control, Vol. 9, pp. 446–475, 1971.Google Scholar
  2. 2.
    Duncan, T. E., andVaraiya, P.,On the Solutions of a Stochastic Control System, I, SIAM Journal on Control, Vol. 9, pp. 354–371, 1971.Google Scholar
  3. 3.
    Duncan, T. E., andVaraiya, P.,On the Solutions of a Stochastic Control System, II, SIAM Journal on Control, Vol. 13, pp. 1077–1092, 1975.Google Scholar
  4. 4.
    Segal, I. E.,Distributions in Hilbert Space and Canonical Systems of Operators, Transactions of the American Mathematical Society, Vol. 88, pp. 12–41, 1958.Google Scholar
  5. 5.
    Gross, L.,Abstract Wiener Spaces, Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, pp. 31–42, University of California Press, Berkeley, California, 1965.Google Scholar
  6. 6.
    Dudley, R. M., Feldman, J., andLe Cam, L.,On Seminorms, Probabilities, and Abstract Wiener Spaces, Annals of Mathematics, Vol. 93, pp. 390–408, 1971.Google Scholar
  7. 7.
    Eels, J.,On the Geometry of Function Spaces, Symposio Internacional de Topologia Algebráica, pp. 303–308, Mexico City, Mexico, 1958.Google Scholar
  8. 8.
    Kobayashi, S., andNomizu, K.,Foundations of Differential Geometry, Vol. 1, John Wiley and Sons (Interscience Publishers), New York, New York, 1963.Google Scholar
  9. 9.
    McKean, H. P.,Brownian Motions on the 3-Dimensional Rotation Group, Memoirs of the College of Science, Kyoto University, Vol. 33, pp. 25–38, 1960.Google Scholar
  10. 10.
    McKean, H. P.,Stochastic Integrals, Academic Press, New York, New York, 1969.Google Scholar
  11. 11.
    Eels, J., andElworthy, K. D.,Wiener Integration on Certain Manifolds, Some Problems in Nonlinear Analysis, Centro Internazionale Matematico Estivo, 1970.Google Scholar
  12. 12.
    Itô, K.,The Brownian Motion and Tensor Fields on Riemannian Manifold, Proceedings of the International Congress of Mathematicians, pp. 536–539, Stockholm, Sweden, 1963.Google Scholar
  13. 13.
    Duncan, T. E.,Some Stochastic Systems on Manifolds, pp. 262–270, Springer-Verlag, Berlin, Germany, 1975.Google Scholar
  14. 14.
    Segal, I. E.,Tensor Algebras over Hilbert Space, I, Transactions of the American Mathematical Society, Vol. 81, pp. 106–134, 1956.Google Scholar
  15. 15.
    Girsanov, I. V.,On Transforming a Certain Class of Stochastic Processes by Absolutely Continuous Substitution of Measures, Theory of Probability and Its Applications, Vol. 5, pp. 285–301, 1960.Google Scholar
  16. 16.
    Benes, V. E.,Existence of Optimal Strategies Based on Specified Information, for a Class of Stochastic Decision Problems, SIAM Journal on Control, Vol. 8, pp. 179–188, 1970.Google Scholar
  17. 17.
    Kunita, H., andWatanabe, S.,On Square Integrable Martingales, Nagoya Journal of Mathematics, Vol. 30, pp. 209–245, 1967.Google Scholar
  18. 18.
    Himmelberg, C. J., andVan Vleck, F. S.,Selection and Implicit Function Theorems for Multifunctions with Souslin Graph, Bulletin de l'Academie Polonaise des Sciences, Vol. 19, pp. 911–916, 1971.Google Scholar
  19. 19.
    Himmelberg, C. J.,Measurable Relations, Fundamenta Mathematicae, Vol. 87, pp. 53–72, 1975.Google Scholar

Copyright information

© Plenum Publishing Corporation 1979

Authors and Affiliations

  • T. E. Duncan
    • 1
  1. 1.Department of MathematicsUniversity of KansasLawrence

Personalised recommendations