Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Linearization and local stability of random dynamical systems

Authors: Igor V. Evstigneev, Sergey A. Pirogov and Klaus R. Schenk-Hoppé
Journal: Proc. Amer. Math. Soc. 139 (2011), 1061-1072
MSC (2010): Primary 37H05, 34F05; Secondary 91G80, 37H15
Published electronically: September 24, 2010
MathSciNet review: 2745656
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The paper examines questions of local asymptotic stability of random dynamical systems. Results concerning stochastic dynamics in general metric spaces, as well as in Banach spaces, are obtained. The results pertaining to Banach spaces are based on the linearization of the systems under study. The general theory is motivated (and illustrated in this paper) by applications in mathematical finance.

References [Enhancements On Off] (What's this?)

  • 1. L. Arnold. Random Dynamical Systems. Springer, 1998. MR 1723992 (2000m:37087)
  • 2. L. Arnold, I. V. Evstigneev and V. M. Gundlach. Convex-valued random dynamical systems: A variational principle for equilibrium states. Random Oper. Stoch. Equ. 7 (1999), 23-38. MR 1677758 (2000j:60078)
  • 3. L. Arnold, V. M. Gundlach and L. Demetrius. Evolutionary formalism for products of positive random matrices. Ann. Appl. Probab. 4 (1994), 859-901. MR 1284989 (95h:28028)
  • 4. A. Carverhill. Flows of stochastic dynamical systems: Ergodic theory. Stochastics 14 (1985), 273-317. MR 805125 (87c:58059)
  • 5. E. A. Coayla-Teran, S.-E. A. Mohammed and P. R. C. Ruffino. Hartman-Grobman theorems along hyperbolic stationary trajectories. Discrete Contin. Dyn. Syst. 17 (2007), 281-292. MR 2257433 (2007j:37084)
  • 6. E. A. Coayla-Teran and P. R. C. Ruffino. Stochastic versions of Hartman-Grobman theorems. Stoch. Dyn. 4 (2004), 571-593. MR 2102754 (2005g:37101)
  • 7. N. D. Cong. Topological classification of linear hyperbolic cocycles. J. Dynam. Differential Equations 8 (1996), 427-467. MR 1412246 (97j:58117)
  • 8. C. Dellacherie and P.-A. Meyer. Probabilities and Potential. North-Holland, 1978. MR 521810 (80b:60004)
  • 9. L. Demetrius and V. M. Gundlach. Evolutionary dynamics in random environments. In: H. Crauel and V. M. Gundlach (eds.), Stochastic Dynamics. Springer, 1999, pp. 371-394. MR 1678507 (99m:58176)
  • 10. I. V. Evstigneev. Positive matrix-valued cocycles over dynamical systems. Uspekhi Matem. Nauk (Russ. Math. Surveys) 29, No. 5 (1974), 219-220. (In Russian) MR 0396906 (53:766)
  • 11. I. V. Evstigneev, T. Hens and K. R. Schenk-Hoppé. Evolutionary stable stock markets. Econom. Theory 27 (2006), 449-468. MR 2212761 (2007h:91087)
  • 12. I. V. Evstigneev, T. Hens and K. R. Schenk-Hoppé. Evolutionary finance. In: T. Hens and K. R. Schenk-Hoppé (eds.), Handbook of Financial Markets: Dynamics and Evolution. North-Holland, 2009, pp. 507-566.
  • 13. I. V. Evstigneev and S. A. Pirogov. A stochastic contraction principle. Random Oper. Stoch. Equ. 15 (2007), 155-162. MR 2337912 (2008h:47115)
  • 14. I. V. Evstigneev and S. A. Pirogov. Stochastic nonlinear Perron-Frobenius theorem. Positivity 14 (2010), 43-57. MR 2596462
  • 15. I. V. Evstigneev and K. R. Schenk-Hoppé. Stochastic equilibria in von Neumann-Gale dynamical systems. Trans. Amer. Math. Soc. 360 (2008), 3345-3364. MR 2379800 (2010b:91135)
  • 16. H. Furstenberg and H. Kesten. Products of random matrices. Ann. Math. Statist. 31 (1960), 457-469. MR 0121828 (22:12558)
  • 17. D. M. Grobman. Homeomorphisms of systems of differential equations. Dokl. Akad. Nauk SSSR 128 (1959), 880-881. MR 0121545 (22:12282)
  • 18. P. Hartman. A lemma in the theory of structural stability of differential equations. Proc. Amer. Math. Soc. 11 (1960), 610-620. MR 0121542 (22:12279)
  • 19. P. Hartman. On local homeomorphisms of Euclidean spaces. Bol. Soc. Mat. Mexicana 5 (1960), 220-241. MR 0141856 (25:5253)
  • 20. K. Khanin and Yu. Kifer. Thermodynamic formalism for random transformations and statistical mechanics. Amer. Math. Soc. Transl. Ser. 2 171 (1996), 107-140. MR 1359097 (96j:58136)
  • 21. Yu. Kifer. Fractal dimensions and random transformations. Trans. Amer. Math. Soc. 348 (1996), 2003-2038. MR 1348865 (96i:28009)
  • 22. Yu. Kifer. Perron-Frobenius theorem, large deviations, and random perturbations in random environments. Math. Z. 222 (1996), 677-698. MR 1406273 (97f:60131)
  • 23. Yu. Kifer. Limit theorems for random transformations and processes in random environments. Trans. Amer. Math. Soc. 350 (1998), 1481-1518. MR 1451607 (98i:60021)
  • 24. Yu. Kifer. Thermodynamic formalism for random transformations revisited. Stoch. Dyn. 8 (2008), 77-102. MR 2399927 (2009a:37066)
  • 25. Yu. Kifer and P.-D. Liu. Random dynamics. In: B. Hasselblatt and A. Katok (eds.), Handbook of Dynamical Systems, Vol. 1B. Elsevier, 2006, pp. 379-499. MR 2186245 (2008a:37002)
  • 26. A. N. Kolmogorov and S. V. Fomin. Elements of the Theory of Functions and Functional Analysis. Graylock, 1957. MR 0085462 (19:44d)
  • 27. Z. Lian and K. Lu. Lyapunov exponents and invariant manifolds for random dynamical systems in a Banach space. Mem. Amer. Math. Soc. 206 (2010), no. 967.
  • 28. L. C. MacLean, E. O. Thorp and W. T. Ziemba (eds.). The Kelly Capital Growth Investment Criterion: Theory and Practice. World Scientific, 2011.
  • 29. S.-E. A. Mohammed and M. K. R. Scheutzow. The stable manifold theorem for stochastic differential equations. Ann. Probab. 27 (1999), 615-652. MR 1698943 (2001e:60121)
  • 30. S.-E. A. Mohammed, T. Zhang and H. Zhao. The stable manifold theorem for semilinear stochastic evolution equations and stochastic partial differential equations. Mem. Amer. Math. Soc. 196 (2008), no. 917. MR 2459571 (2010b:60175)
  • 31. G. Ochs and V. I. Oseledets. Topological fixed point theorems do not hold for random dynamical systems. J. Dynam. Differential Equations 11 (1999), 583-593. MR 1725412 (2000i:37075)
  • 32. V. I. Oseledec. A multiplicative ergodic theorem: Characteristic Ljapunov exponents of dynamical systems. Trans. Moscow Math. Soc. 19 (1968), 197-231. MR 0240280 (39:1629)
  • 33. S. A. Pirogov and Ya. G. Sinai. Phase diagrams of classical lattice systems, I and II, Theor. Math. Phys. 25 (1975), 1185-1192; III, Theor. Math. Phys. 26 (1976), 39-49. MR 0676316 (58:32680); MR 0676499 (58:32712)
  • 34. D. Ruelle. Ergodic theory of differentiable dynamical systems. Publ. Math. Inst. Hautes Études Sci. 50 (1979), 27-58. MR 556581 (81f:58031)
  • 35. D. Ruelle. Characteristic exponents and invariant manifolds in Hilbert space. Ann. of Math. (2) 115 (1982), 243-290. MR 647807 (83j:58097)
  • 36. B. Schmalfuss. A random fixed point theorem and the random graph transformation. J. Math. Anal. Appl. 225 (1998), 91-113. MR 1639297 (99i:47118)
  • 37. Ya. G. Sinai. Gibbs measures in ergodic theory. Uspekhi Matem. Nauk (Russ. Math. Surveys) 27 (1972), 21-69. MR 0399421 (53:3265)
  • 38. T. Wanner. Linearization of random dynamical systems. In: C. John, U. Kirchgraber and H. O. Walther (eds.), Dynamics Reported, Vol. 4. Springer, 1994, pp. 203-269. MR 1346499 (96m:34084)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 37H05, 34F05, 91G80, 37H15

Retrieve articles in all journals with MSC (2010): 37H05, 34F05, 91G80, 37H15

Additional Information

Igor V. Evstigneev
Affiliation: Department of Economics, University of Manchester, Manchester M13 9PL, United Kingdom

Sergey A. Pirogov
Affiliation: Institute for Information Transmission Problems, Academy of Sciences of Russia, GSP-4, Moscow, 101447, Russia

Klaus R. Schenk-Hoppé
Affiliation: School of Mathematics and Leeds University Business School, University of Leeds, Leeds LS2 9JT, United Kingdom

Keywords: Local stability, linearization, random fixed points, random dynamical systems, mathematical finance.
Received by editor(s): March 29, 2010
Published electronically: September 24, 2010
Additional Notes: The authors gratefully acknowledge financial support from the Swiss National Center of Competence in Research “Financial Valuation and Risk Management” (project “Behavioural and Evolutionary Finance”) and from the Finance Market Fund, Norway (projects “Stochastic Dynamics of Financial Markets” and “Stability of Financial Markets: An Evolutionary Approach”).
Communicated by: Yingfei Yi
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society