Skip to main content

Stochastic Differential Systems With Memory: Theory, Examples and Applications

  • Conference paper
Stochastic Analysis and Related Topics VI

Part of the book series: Progress in Probability ((PRPR,volume 42))

Abstract

The purpose of this article is to introduce the reader to certain aspects of stochastic differential systems whose evolution depends on the past history of the state.

Research supported in part by NSF Grants DMS-9206785 and DMS-9503702.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Ahmed, T.A., M.Sc. Thesis, University of Khartoum, (1983).

    Google Scholar 

  2. Arnold, L., Kliemann, W. and Oeljeklaus, E. Lyapunov exponents of linear stochastic systems, in Lyapunov Exponents, Springer Lecture Notes in Mathematics, 1186 (1989), 85–125.

    Book  Google Scholar 

  3. Arnold, L., Oeljeklaus, E. and Pardoux, E., Almost sure and moment stability for linear Itô equations, in Lyapunov Exponents, Springer Lecture Notes in Mathematics, 1186 (ed. L. Arnold and V. Wihstutz) (1986), 129–159.

    Chapter  Google Scholar 

  4. Baxendale, P. H., Moment stability and large deviations for linear stochastic differential equations, in Ikeda, N. (ed.) Proceedings of the Taniguchi Symposium on Probabilistic Methods in Mathematical Physics, Katata and Kyoto (1985), 31–54, Tokyo: Kinokuniya (1987).

    Google Scholar 

  5. Bailey, H. R. and Williams, M. Z., Some results on the differential difference equation \( x'(t) = \sum\nolimits_{{i = 0}}^N {{A_i}x\left( {t - {T_i}} \right)} \), J. Math. Anal Appl. 15 (1966), 569–587.

    Article  MathSciNet  MATH  Google Scholar 

  6. Bell, D. R., Degenerate Stochastic Differential Equations and Hypoellipticity, Pitman Monographs and Surveys in Pure and Applied Mathematics, Vol. 79, Longman, Essex, 1995.

    MATH  Google Scholar 

  7. Bell, D. R. and Mohammed, S.-E. A., The Malliavin calculus and stochastic delay equations, J. Fund Anal. 99, No. 1 (1991), 75–99.

    Article  MathSciNet  MATH  Google Scholar 

  8. Bell, D.R. and Mohammed, S.-E. A., An extension of Hörmander’s theorem for infinitely degenerate parabolic operators, Duke Math. Journal 78 (1995), no. 3, 453–475.

    Article  MathSciNet  MATH  Google Scholar 

  9. Bell, D. R. and Mohammed, S.-E. A., Opérateurs paraboliques hypoelliptiques avec dégénérescences exponentielles, C.R. Acad. Sci. Paris, t. 317, Série I, (1993), 1059–1064.

    MathSciNet  MATH  Google Scholar 

  10. Bell, D. R. and Mohammed, S.-E. A., Smooth densities for degenerate stochastic delay equations with hereditary drift, Ann. Prob. 23, no. 4, (1995) 1875–1894.

    Article  MathSciNet  MATH  Google Scholar 

  11. Bell, D. R. and Mohammed, S.-E. A., On the solution of stochastic ordinary differential equations via small delays, Stochastics and Stochastics Rep. 28 (1989), no. 4, 293–299

    MathSciNet  MATH  Google Scholar 

  12. Çinlar, E., Jacod, J., Protter, P. and Sharpe, M., Semimartingales and Markov processes, Z. Wahrsch. Verw. Gebiete, 54 (1980), 161–219.

    Article  MathSciNet  MATH  Google Scholar 

  13. Doleans-Dade, C., On the existence and unicity of solutions of stochastic integral equations, Z. Wahrsch. Verw. Gebiete, 36 (1976), 93–101.

    Article  MathSciNet  MATH  Google Scholar 

  14. Doss, H., Liens entre équations différentielles stochastiques et ordinaires, Ann. Inst. Henri Poincaré, Vol. XIII, no. 2 (1977), 99–125.

    MathSciNet  Google Scholar 

  15. Dudley, R. M., The sizes of compact subsets of Hilbert space and continuity of Gaussian processes, J. Functional Analysis, 1, (1967), 290–330.

    Article  MathSciNet  MATH  Google Scholar 

  16. Dynkin, E. B., Markov Processes, Vols I, II, Springer-Verlag, Berlin (1965).

    MATH  Google Scholar 

  17. Delfour, M. C. and Mitter, S. K., Hereditary differential systems with constant delays. I. General case. J. Differential Equations 12 (1972), 213–235; erratum, ibid. 14 (1973), 397.

    Article  MathSciNet  MATH  Google Scholar 

  18. Dunford and Schwartz, Linear Operators, Part I: General Theory, Interscience Publishers, New York (1958).

    Google Scholar 

  19. Flandoli, F. and Schaumlöffel, K.-U., Stochastic parabolic equations in bounded domains: Random evolution operator and Lyapunov exponents, Stochastics and Stochastic Reports, 29, 4 (1990), 461–485.

    MATH  Google Scholar 

  20. Frampton, J. and Tromba, A. J. On the classification of spaces of Hölder continuous functions, J. Functional Analysis, 10 (1972), 336–345

    Article  MathSciNet  MATH  Google Scholar 

  21. Friedman, A., Stochastic Differential Equations and Applications, Vol. 1, Academic Press, New York, San Francisco, London (1975).

    MATH  Google Scholar 

  22. Garsia, A. M., Rodemich, E. and Rumsey, H., Jr., A real variable lemma and the continuity of paths of some Gaussian processes, Indiana Univ. Math. J., 20 (1970/1971), 565–578.

    Article  MathSciNet  MATH  Google Scholar 

  23. Gihman, I. I. and Skorohod, A. V., Stochastic Differential Equations, Springer-Verlag, New York (1973).

    Google Scholar 

  24. Hale, J. K., Theory of Functional Differential Equations, Springer-Verlag, New York, Heidelberg, Berlin, (1977).

    Book  MATH  Google Scholar 

  25. Has’minskiĭ, R. Z., Stochastic Stability of Differential Equations, Sijthoff and Noordhoff (1980).

    Book  Google Scholar 

  26. Itô, K. and Nisio, M., On stationary solutions of a stochastic differential equation, J. Math. Kyoto University, 4–1 (1964), 1–75.

    Google Scholar 

  27. Kubo, R., The fluctuation-dissipation theorem and Brownian motion, in Many-Body Theory edited by R. Kubo, Syokabo and Benjamin (1966), 1–16.

    Google Scholar 

  28. Kunita, H., Stochastic Flows and Stochastic Differential Equations, Cambridge University Press, Cambridge, New York, Melbourne, Sydney (1990).

    MATH  Google Scholar 

  29. Kushner, H. J., On the stability of processes defined by stochastic differential-difference equations, J. Differential Equations, 4, (1968), 424–443.

    Article  MathSciNet  MATH  Google Scholar 

  30. Kolmanovskii, V. B. and Nosov, V. R., Stability of Functional Differential Equations, Academic Press, London, Orlando (1986).

    MATH  Google Scholar 

  31. Kusuoka, S. and Stroock, D. W., Applications of the Malliavin calculus, I, Taniguchi Sympos. SA Katata (1982), 271–306.

    Google Scholar 

  32. Kusuoka, S. and Stroock, D. W., Applications of the Malliavin calculus, II, J. Fac. Sei. Univ. Tokyo, Sect. 1A Math., 32, No. 1 (1985), 1–76.

    MathSciNet  MATH  Google Scholar 

  33. Lenhart, S. M. and Travis, C. C, Global stability of a biological model with time delay, Proc. Amer. Math. Soc., 96, no. 1, (1986),75–78.

    Article  MathSciNet  MATH  Google Scholar 

  34. Lipster, R. S. and Shiryayev, A. N. Statistics of Random Processes, Springer, Berlin (1977).

    Google Scholar 

  35. Mao, X.R., Exponential Stability of Stochastic Differential Equations, Pure and Applied Mathematics, Marcel Dekker, New York-Basel-Hong Kong (1994).

    MATH  Google Scholar 

  36. Metivier, M. Semimartingales, a Course on Stochastic Processes, Walter de Gruyter, Berlin-New York (1982).

    Book  MATH  Google Scholar 

  37. Mohammed, S.-E. A. Stochastic Functional Differential Equations, Research Notes in Mathematics, 99, Pitman Advanced Publishing Program, Boston, London, Melbourne (1984).

    MATH  Google Scholar 

  38. Mohammed, S.-E. A., Non-linear flows for linear stochastic delay equations, Stochastics, 17, 3 (1986), 207–212.

    Article  MathSciNet  MATH  Google Scholar 

  39. Mohammed, S.-E. A., The Lyapunov spectrum and stable manifolds for stochastic linear delay equations, Stochastics and Stochastic Reports, 29 (1990), 89–131.

    MathSciNet  MATH  Google Scholar 

  40. Mohammed, S.-E. A., Lyapunov exponents and stochastic flows of linear and affine hereditary systems, (Survey article), in Diffusion Processes and Related Problems in Analysis, Volume II, edited by M. Pinsky and V. Wihstutz, Birkhäuser (1992), 141–169.

    Chapter  Google Scholar 

  41. Marcus, M. and Mizel, V. J., Stochastic functional differential equations modeling materials with selective recall, Stochastics 25 (1988), no. 4, 195–232

    Article  MathSciNet  MATH  Google Scholar 

  42. Metivier, M. and Pellaumail, J. Stochastic Integration, Academic Press, London-New York (1980).

    MATH  Google Scholar 

  43. Mizel, V. J. and Trutzer, V., Stochastic hereditary equations: existence and asymptotic stability, Journal of Integral Equations 7 (1984), 1–72.

    Article  MathSciNet  MATH  Google Scholar 

  44. Mohammed, S.-E. A. and Scheutzow, M. K. R., Lyapunov exponents of linear stochastic functional differential equations driven by semimartingales. Part I: The multiplicative ergodic theory, Ann. Inst. Henri Poincaré, Probabilités et Statistiques, vol. 32, (1996), 69–105.

    MathSciNet  MATH  Google Scholar 

  45. Mohammed, S.-E. A. and Scheutzow, M. K. R., Lyapunov exponents of linear stochastic functional differential equations driven by semimartingales. Part II: Examples and case studies, Ann. Prob., 6, 3 (1997), 1210–1240.

    MathSciNet  Google Scholar 

  46. Mohammed, S.-E. A. and Scheutzow, M. K. R., Lyapunov exponents and stationary solutions for affine stochastic delay equations, Stochastics and Stochastic Reports, Vol. 29, No. 2 (1990), 259–283.

    MathSciNet  MATH  Google Scholar 

  47. Pardoux, E. and Wihstutz, V., Lyapunov exponent and rotation number of two-dimensional stochastic systems with small diffusion, SIAM J. Applied Math., 48, (1988), 442–457.

    Article  MathSciNet  MATH  Google Scholar 

  48. Pinsky, M. and Wihstutz, V., Lyapunov exponents of nilpotent Itô systems, Stochastics, 25, (1988), 43–57.

    Article  MathSciNet  MATH  Google Scholar 

  49. Protter, Ph. E., On the existence, uniqueness and explosion of solutions of systems of stochastic integral equations, Ann. Prob. vol. 5, (1977), 243–261.

    Article  MathSciNet  MATH  Google Scholar 

  50. Protter, Ph. E., Semimartingales and measure-preserving flows, Ann. Inst. Henri Poincaré, Probabilités et Statistiques, vol. 22, (1986), 127–147.

    MathSciNet  MATH  Google Scholar 

  51. Ruelle, D., Characteristic exponents and invariant manifolds in Hilbert space, Annals of Mathematics, 115 (1982), 243–290.

    Article  MathSciNet  MATH  Google Scholar 

  52. Scheutzow, M. K. R., Stationary and Periodic Stochastic Differential Systems: A study of Qualitative Changes with Respect to the Noise Level and Asymptotics, Habiltationsschrift, University of Kaiserslautern, Germany (1988).

    Google Scholar 

  53. Schwartz, L., Radon Measures on Arbitrary Topological Spaces and Cylindrical measures, Tata Institute of Fundamental Research, Oxford University Press, (1973).

    MATH  Google Scholar 

  54. Skorohod, A. V. Random Linear Operators, D. Reidel Publishing Company (1984).

    Book  Google Scholar 

  55. Stroock, D. W., The Malliavin calculus, a functional analytic approach, J. Functional Anaysis, 44 (1981), 212–257.

    Article  MathSciNet  MATH  Google Scholar 

  56. Sussman, H. J., On the gap between deterministic and stochastic ordinary differential equations, Ann. Prob., 6, 1 (1978), 19–41.

    Article  Google Scholar 

  57. de Sam Lazaro, J. and Meyer, P. A., Méthodes de martingales et théorie des flots, Z. Wahrsch. Verw. Gebiete, Vol. 18, (1971), 116–140.

    Article  MATH  Google Scholar 

  58. de Sam Lazaro, J. and Meyer, P. A., Questions de théorie des flots, Séminaire de Probab. IX, Springer Lecture Notes in Mathematics, 465 (1975), 1–96.

    Google Scholar 

  59. Tromba, A. J., On the isometries of spaces of Hölder continuous functions, Studia Math. 57, no. 3 (1976), 199–208.

    MathSciNet  MATH  Google Scholar 

Download references

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1998 Springer Science+Business Media New York

About this paper

Cite this paper

Mohammed, SE.A. (1998). Stochastic Differential Systems With Memory: Theory, Examples and Applications. In: Decreusefond, L., Øksendal, B., Gjerde, J., Üstünel, A.S. (eds) Stochastic Analysis and Related Topics VI. Progress in Probability, vol 42. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-2022-0_1

Download citation

  • DOI: https://doi.org/10.1007/978-1-4612-2022-0_1

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-1-4612-7385-1

  • Online ISBN: 978-1-4612-2022-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics