Skip to main content

Almost sure and moment stability for linear ito equations

Part II: Linear Stochastic Systems. Stability Theory

Part of the Lecture Notes in Mathematics book series (LNM,volume 1186)

Abstract

The asymptotic behavior of the linear stochastic differential equation in Rd

$$dx = Ax dt + \mathop \Sigma \limits_{i = 1}^m B_i x o dW_i (t), x(O) = x_O \ne O,$$

is studied. It is known (see [2]) in these Proceedings) that the projection of the solution x(t;xo) onto the unit sphere has a unique invariant probability, while

$$\lambda = \mathop {\lim }\limits_{t \to \infty } \tfrac{1}{t} \log |x(t;x_O )|$$

exists a.s. and is essentially independent of chance and of xo. Here we prove that

$$g(p) = \mathop {\lim }\limits_{t \to \infty } \tfrac{1}{t} \log E|x(t;x_O )|^P , p \in R,$$

exists and is independent of xo. Further, g: R → R is convex and analytic with g(p)/p increasing (to γ, say) with g(O)=O and g' (O)=λ. The cases γ<∞ and γ=∞ are characterized. This answers the question of when sample stability (λ<O) implies moment stability (g(p)<O) for all p>O.

In case trace A = trace Bi=O for all i we have g(−d)=O, this enabling us to characterize the cases λ=O and λ>O by a simple criterion.

Keywords

  • Lyapunov Exponent
  • Stochastic Differential Equation
  • Fundamental Matrix
  • Lyapunov Spectrum
  • Linear Stochastic System

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, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   34.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   46.00
Price excludes VAT (Canada)
  • Compact, lightweight 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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Arnold, L.: A formula connecting sample and moment stability of linear stochastic systems. SIAM J. Appl. Math. 44(1984), 793–802.

    CrossRef  MathSciNet  MATH  Google Scholar 

  2. Arnold, L., Kliemann, W., and Oeljeklaus, E.: Lyapunov exponents of linear stochastic systems. These Proceedings.

    Google Scholar 

  3. Bhattacharya, R.N.: On the functional central limit theorem and the law of the iterated logarithm for Markov processes. Z. Wahrscheinlichkeitstheorie verw. Gebiete 60(1982), 185–201.

    CrossRef  MathSciNet  MATH  Google Scholar 

  4. Bony, J.-M.: Principe du maximum, inégalité de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés. Ann. Inst. Fourier (Grenoble) 19(1969), 277–304.

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. Borel, A.: Les bouts des éspaces homogènes de groupes de Lie. Ann. of Math. 58(1953), 443–457.

    CrossRef  MathSciNet  MATH  Google Scholar 

  6. Carverhill, A.: Flows of stochastic dynamical systems: ergodic theory. Stochastics 14(1985), 273–317.

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. Chappell, M.: Bounds for average Lyapunov exponents of gradient stochastic systems. These Proceedings.

    Google Scholar 

  8. Greiner, G.: Spektrum und Asymptotic stark stetiger Halbgruppen positiver Operatoren. Sitzungsber. Heidelberger Akad. d. Wiss. (math.-naturw. Klasse), 3. Abhandlung (1982).

    Google Scholar 

  9. Ichihara, K., and Kunita, H.: A classification of the second order degenerate elliptic operators and its probabilistic characterization. Z. Wahrscheinlichkeitstheorie verw. Gebiete 30(1974), 235–254, and 39(1977), 81–84 (Supplements).

    CrossRef  MathSciNet  MATH  Google Scholar 

  10. Ikeda, N., and Watanabe, S.: Stochastic differential equations and diffusion processes. Amsterdam: North Holland 1981.

    MATH  Google Scholar 

  11. Kato, T.: Perturbation theory for linear operators. Berlin-Heidelberg-New York: Springer 1980.

    MATH  Google Scholar 

  12. Kliemann, W.: Recurrence and invariant measures for degenerate diffusions. Annals of Prob. (to appear).

    Google Scholar 

  13. Kunita, H.: Diffusion processes and control systems. Lecture Notes, University of Paris VI, 1974.

    Google Scholar 

  14. Kunita, H.: Supports of diffusion processes and controllability problems. Proceed. Intern. Symp. Stochastic Diff. Equs., Kyoto 1976, 163–185. New York: Wiley 1978.

    MATH  Google Scholar 

  15. Kunita, H.: Stochastic differential equations and stochastic flows of diffeomorphisms. École d′ Été de Probabilités de Saint-Flour XII — 1982. Lecture Notes in Mathematics 1097, 143–303. Berlin-Heidelberg-New York-Tokyo: Springer 1984.

    CrossRef  Google Scholar 

  16. Ledrappier, F.: Quelques proprietés des éxposants caracteristiques. École d' Été de Probabilités de Saint-Flour XII-1982. Lecture Notes in Mathematics 1097, 305–396. Berlin-Heidelberg-New York-Tokyo: Springer 1984.

    CrossRef  Google Scholar 

  17. Oseledec, V.I.: A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems. Trans. Moscow Math. Soc. 19(1968), 197–231.

    MathSciNet  Google Scholar 

  18. Pardoux, E., and Protter, P.: Two-sided stochastic integral and calculus (to appear).

    Google Scholar 

  19. Pignol, M.: Stabilité stochastique des pales d' helicoptère. Thèse de troisieme cycle, Université de Provence. Marseille 1985.

    Google Scholar 

  20. Ruelle, D.: Ergodic theory of differentiable dynamical systems. Publ. math. de l'IHES 50(1979), 27–58.

    CrossRef  MathSciNet  MATH  Google Scholar 

  21. Sussmann, H., and Jurdjevic, V.: Controllability of nonlinear systems. J. Diff. Equs. 12(1972), 95–116.

    CrossRef  MathSciNet  MATH  Google Scholar 

  22. Willems, J.L., and Willems, J.C.: Robust stabilization of uncertain systems. SIAM J. Control and Optim. 21(1983), 352–374.

    CrossRef  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1986 Springer-Verlag

About this paper

Cite this paper

Arnold, L., Oeljeklaus, E., Pardoux, E. (1986). Almost sure and moment stability for linear ito equations. In: Arnold, L., Wihstutz, V. (eds) Lyapunov Exponents. Lecture Notes in Mathematics, vol 1186. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0076837

Download citation

  • DOI: https://doi.org/10.1007/BFb0076837

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-16458-6

  • Online ISBN: 978-3-540-39795-3

  • eBook Packages: Springer Book Archive