Stochastic analysis on differential structures

  • M. M. Rao
Chapter
Part of the Mathematics and Its Applications book series (MAIA, volume 342)

Abstract

In most of the preceding work on stochastic integral and differential analysis, the study was based on the real line as its range space as well as its parameter set. However, modern developments demand that we consider the analogous theory on differential structures, more general than the real line. In this final chapter we first discuss the conformal properties of martingales with values in the complex plane and then devote Section 2 for a formulation and an extension of the work on certain (differential or smooth) manifolds, leading to Malliavin’s approach to the hypoellipticity problem. The study continues in Section 3 by discussing semimartingales with ℝn, n ≥ 2, as the parameter set. There are substantial difficulties in this extension due to a lack of linear ordering. We observe that, even here, the L2,2-boundedness principle relative to a (σ-finite) measure is at work and corresponding stochastic integrals can be defined. Finally we indicate in Section 4 how one considers stochastic partial differential equations in this context. Then some complements to the preceding work are given, as in earlier chapters, in the last section to conclude this monograph.

Keywords

Brownian Motion Stochastic Differential Equation Stochastic Analysis Stochastic Partial Differential Equation Preceding Work 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. [1]
    Getoor, R. M., and Sharpe, M. J. “Conformal martingales.” Invent. Math. 16 (1972) 271–308.MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    Föllmer, H. “Stochastic holomorphy.” Math. Ann. 207 (1974) 245–255.MathSciNetCrossRefGoogle Scholar
  3. [1]
    Hörmander, L. “Hypoelliptic second order differential equations.” Acta Math. 119 (1967) 147–171.MathSciNetMATHCrossRefGoogle Scholar
  4. [1]
    Malliavin, P. “Stochastic calculus of variation and hypoelliptic operators.” Proc. Intern. Symp. Stoch. Diff. Eq., Kyoto, (1978) 195–263.Google Scholar
  5. [2]
    Bismut, J.-M. Large Deviations and the Malliavin Calculus. Birkhauser, 1984.Google Scholar
  6. [1]
    Stroock, D. W. “Malliavin calculus: a functional analytic approach.” J. Funct. Anal. 44 (1981) 212–257.MathSciNetMATHCrossRefGoogle Scholar
  7. [4]
    Schwartz, L. Semi-Martingales sur des Variétés, et Martingales Conformes sur des Variétés Analytiques Complexes. Lect. Notes Math. 780 (1980) 132 pp.Google Scholar
  8. [7]
    Meyer, P. A. “Geometrie differential stochastiques.” Astérisque, 131(1985) 107113.Google Scholar
  9. [1]
    Auslander, L., and MacKenzie, R. E. Introduction to Differential Manifolds. McGraw-Hill, New York, 1963.Google Scholar
  10. [1]
    Flanders, H. Differential Forms with Applications to Physical Sciences. Dover, New York, 1989.Google Scholar
  11. [1]
    Guggenheimer, H. W. Differential Geometry. Dover, New York, 1977.Google Scholar
  12. [1]
    Goldberg, S. I. Curvature and Homology. Dover, New York, 1982.Google Scholar
  13. [1]
    Postnikov, M. M. The Variational Theory of Geodesics. ( Translation) Dover, New York, 1983.Google Scholar
  14. [1]
    Emery, M. Stochastic Calculus in Manifolds. Springer-Verlag, New York, 1989.MATHCrossRefGoogle Scholar
  15. [1]
    Bell, D. R. The Malliavin Calculus. Pitman Math. Mono. 34 London, 1987.Google Scholar
  16. [1]
    Ikeda, N., and Watanabe, S. Stochastic Differential Equations and Diffusion Processes, North-Holland, Amsterdam, (2nd ed.) 1989.Google Scholar
  17. [1]
    Belopol’skaya, Ya. I., and Dalecky, Yu. L. Stochastic Equations and Differential Geometry. Kluwer Acad. Publ. Boston, MA 1990.Google Scholar
  18. [1]
    Cairoli, R. “Une inéqualité pour martingales à indices multiples et ses applications.” Lect. Notes Math. 124 (1970) 49–80.Google Scholar
  19. [1]
    Cairoli, R., and Walsh, J. B. “Stochastic integrals in the plane.” Acta Math. 134(1975) 111183.Google Scholar
  20. [2]
    Walsh, J. B. “Martingales with a multidimensional parameter and stochastic integrals in the plane.” Lect. Notes Math. 1215 (1986) 329–491.CrossRefGoogle Scholar
  21. [1]
    Brennan, M. D. “Planar semimartingales.” J. Multivar. Anal. 9 (1979) 465–486.MathSciNetMATHCrossRefGoogle Scholar
  22. [1]
    Kailath, T., and Zakai, M. “Absolute continuity and Radon-Nikodÿm derivatives for certain measures relative to Wiener measure.” Ann. Math. Statist. 42 (1971) 130–140.MathSciNetMATHCrossRefGoogle Scholar
  23. [1]
    Dozzi, M. Stochastic Processes with a Multidimensional Parameter. Res.Notes Math. Pitman, London, 1989.Google Scholar
  24. [1]
    Dubins, L. E. “Conditional probability distributions in the wide sense.” Proc. Am. Math. Soc. 8 (1957) 1088–1092.MathSciNetCrossRefGoogle Scholar
  25. [1]
    Hürzeler, H. E. “Stochastic integration on partially ordered sets.” J. Multivar. Anal. 17 (1985) 279–303.MATHCrossRefGoogle Scholar
  26. [2]
    Yeh, J. “Two parameter stochastic differential equations.” in Real and Stochastic Analysis, Wiley, New York, (1986) 249–344. Ylinen, K.Google Scholar
  27. [1]
    Yeh, J. “On vector bimeasures.” Ann. Mat. Pura Appl. 117 (4) (1978) 115–138.MathSciNetGoogle Scholar
  28. [1]
    Green, M. L. Multiparameter Semimartingale Integrals and Boundedness Principles. Ph. D. thesis, Univ. Calif., Riverside, 1995.Google Scholar
  29. [1]
    Kunita, H. Stochastic Flows and Stochastic Differential Equations. Camb. Univ. Press, London, 1990.Google Scholar
  30. [1]
    Freidlin, M. Functional Integration and Partial Differential Equations. Princeton Univ. Press, Princeton, NJ, 1985.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1995

Authors and Affiliations

  • M. M. Rao
    • 1
  1. 1.University of CaliforniaRiversideUSA

Personalised recommendations