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
DOI:
https://doi.org/10.1090/S0002-9939-2010-10647-0
Published electronically:
September 24, 2010
MathSciNet review:
2745656
Full-text PDF Free Access
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.
- 1. Ludwig Arnold, Random dynamical systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. MR 1723992
- 2. L. Arnold, I. V. Evstigneev, and V. M. Gundlach, Convex-valued random dynamical systems: a variational principle for equilibrium states, Random Oper. Stochastic Equations 7 (1999), no. 1, 23–38. MR 1677758, https://doi.org/10.1515/rose.1999.7.1.23
- 3. Ludwig Arnold, Volker Matthias Gundlach, and Lloyd Demetrius, Evolutionary formalism for products of positive random matrices, Ann. Appl. Probab. 4 (1994), no. 3, 859–901. MR 1284989
- 4. Andrew Carverhill, Flows of stochastic dynamical systems: ergodic theory, Stochastics 14 (1985), no. 4, 273–317. MR 805125, https://doi.org/10.1080/17442508508833343
- 5. Edson A. Coayla-Teran, Salah-Eldin A. Mohammed, and Paulo Régis C. Ruffino, Hartman-Grobman theorems along hyperbolic stationary trajectories, Discrete Contin. Dyn. Syst. 17 (2007), no. 2, 281–292. MR 2257433, https://doi.org/10.3934/dcds.2007.17.281
- 6. Edson A. Coayla-Teran and Paulo R. C. Ruffino, Stochastic versions of Hartman-Grobman theorems, Stoch. Dyn. 4 (2004), no. 4, 571–593. MR 2102754, https://doi.org/10.1142/S0219493704001206
- 7. Nguyen Dinh Cong, Topological classification of linear hyperbolic cocycles, J. Dynam. Differential Equations 8 (1996), no. 3, 427–467. MR 1412246, https://doi.org/10.1007/BF02218762
- 8. Claude Dellacherie and Paul-André Meyer, Probabilities and potential, North-Holland Mathematics Studies, vol. 29, North-Holland Publishing Co., Amsterdam-New York; North-Holland Publishing Co., Amsterdam-New York, 1978. MR 521810
- 9. Lloyd Demetrius and Volker Matthias Gundlach, Evolutionary dynamics in random environments, Stochastic dynamics (Bremen, 1997) Springer, New York, 1999, pp. 371–394. MR 1678507, https://doi.org/10.1007/0-387-22655-9_16
- 10. I. V. Evstigneev, Positive matrix-valued cocycles over dynamical systems, Uspehi Mat. Nauk 29 (1974), no. 5(179), 219–220 (Russian). MR 0396906
- 11. Igor V. Evstigneev, Thorsten Hens, and Klaus Reiner Schenk-Hoppé, Evolutionary stable stock markets, Econom. Theory 27 (2006), no. 2, 449–468. MR 2212761, https://doi.org/10.1007/s00199-005-0607-8
- 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), no. 2, 155–162. MR 2337912, https://doi.org/10.1515/rose.2007.010
- 14. Igor V. Evstigneev and Sergey A. Pirogov, Stochastic nonlinear Perron-Frobenius theorem, Positivity 14 (2010), no. 1, 43–57. MR 2596462, https://doi.org/10.1007/s11117-008-0003-2
- 15. Igor V. Evstigneev and Klaus Reiner Schenk-Hoppé, Stochastic equilibria in von Neumann-Gale dynamical systems, Trans. Amer. Math. Soc. 360 (2008), no. 6, 3345–3364. MR 2379800, https://doi.org/10.1090/S0002-9947-08-04445-0
- 16. H. Furstenberg and H. Kesten, Products of random matrices, Ann. Math. Statist. 31 (1960), 457–469. MR 121828, https://doi.org/10.1214/aoms/1177705909
- 17. D. M. Grobman, Homeomorphism of systems of differential equations, Dokl. Akad. Nauk SSSR 128 (1959), 880–881 (Russian). MR 0121545
- 18. Philip Hartman, A lemma in the theory of structural stability of differential equations, Proc. Amer. Math. Soc. 11 (1960), 610–620. MR 121542, https://doi.org/10.1090/S0002-9939-1960-0121542-7
- 19. Philip Hartman, On local homeomorphisms of Euclidean spaces, Bol. Soc. Mat. Mexicana (2) 5 (1960), 220–241. MR 141856
- 20. K. Khanin and Y. Kifer, Thermodynamic formalism for random transformations and statistical mechanics, Sinaĭ’s Moscow Seminar on Dynamical Systems, Amer. Math. Soc. Transl. Ser. 2, vol. 171, Amer. Math. Soc., Providence, RI, 1996, pp. 107–140. MR 1359097, https://doi.org/10.1090/trans2/171/10
- 21. Yuri Kifer, Fractal dimensions and random transformations, Trans. Amer. Math. Soc. 348 (1996), no. 5, 2003–2038. MR 1348865, https://doi.org/10.1090/S0002-9947-96-01608-X
- 22. Yuri Kifer, Perron-Frobenius theorem, large deviations, and random perturbations in random environments, Math. Z. 222 (1996), no. 4, 677–698. MR 1406273, https://doi.org/10.1007/PL00004551
- 23. Yuri Kifer, Limit theorems for random transformations and processes in random environments, Trans. Amer. Math. Soc. 350 (1998), no. 4, 1481–1518. MR 1451607, https://doi.org/10.1090/S0002-9947-98-02068-6
- 24. Yuri Kifer, Thermodynamic formalism for random transformations revisited, Stoch. Dyn. 8 (2008), no. 1, 77–102. MR 2399927, https://doi.org/10.1142/S0219493708002238
- 25. Yuri Kifer and Pei-Dong Liu, Random dynamics, Handbook of dynamical systems. Vol. 1B, Elsevier B. V., Amsterdam, 2006, pp. 379–499. MR 2186245, https://doi.org/10.1016/S1874-575X(06)80030-5
- 26. A. N. Kolmogorov and S. V. Fomin, Elements of the theory of functions and functional analysis. Vol. 1. Metric and normed spaces, Graylock Press, Rochester, N. Y., 1957. Translated from the first Russian edition by Leo F. Boron. MR 0085462
- 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. Salah-Eldin A. Mohammed and Michael K. R. Scheutzow, The stable manifold theorem for stochastic differential equations, Ann. Probab. 27 (1999), no. 2, 615–652. MR 1698943, https://doi.org/10.1214/aop/1022677380
- 30. Salah-Eldin A. Mohammed, Tusheng Zhang, and Huaizhong Zhao, The stable manifold theorem for semilinear stochastic evolution equations and stochastic partial differential equations, Mem. Amer. Math. Soc. 196 (2008), no. 917, vi+105. MR 2459571, https://doi.org/10.1090/memo/0917
- 31. Gunter Ochs and Valery I. Oseledets, Topological fixed point theorems do not hold for random dynamical systems, J. Dynam. Differential Equations 11 (1999), no. 4, 583–593. MR 1725412, https://doi.org/10.1023/A:1022670227876
- 32. V. I. Oseledec, A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Trudy Moskov. Mat. Obšč. 19 (1968), 179–210 (Russian). MR 0240280
- 33.
S.
A. Pirogov and Ja.
G. Sinaĭ, Phase diagrams of classical lattice systems,
Teoret. Mat. Fiz. 25 (1975), no. 3, 358–369
(Russian, with English summary). MR
676316
S. A. Pirogov and Ja. G. Sinaĭ, Phase diagrams of classical lattice systems. (Continuation), Teoret. Mat. Fiz. 26 (1976), no. 1, 61–76 (Russian). MR 676499 - 34. David Ruelle, Ergodic theory of differentiable dynamical systems, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 27–58. MR 556581
- 35. David Ruelle, Characteristic exponents and invariant manifolds in Hilbert space, Ann. of Math. (2) 115 (1982), no. 2, 243–290. MR 647807, https://doi.org/10.2307/1971392
- 36. Björn Schmalfuss, A random fixed point theorem and the random graph transformation, J. Math. Anal. Appl. 225 (1998), no. 1, 91–113. MR 1639297, https://doi.org/10.1006/jmaa.1998.6008
- 37. Ja. G. Sinaĭ, Gibbs measures in ergodic theory, Uspehi Mat. Nauk 27 (1972), no. 4(166), 21–64 (Russian). MR 0399421
- 38. Thomas Wanner, Linearization of random dynamical systems, Dynamics reported, Dynam. Report. Expositions Dynam. Systems (N.S.), vol. 4, Springer, Berlin, 1995, pp. 203–269. MR 1346499
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
Email:
igor.evstigneev@manchester.ac.uk
Sergey A. Pirogov
Affiliation:
Institute for Information Transmission Problems, Academy of Sciences of Russia, GSP-4, Moscow, 101447, Russia
Email:
pirogov@mail.ru
Klaus R. Schenk-Hoppé
Affiliation:
School of Mathematics and Leeds University Business School, University of Leeds, Leeds LS2 9JT, United Kingdom
Email:
k.r.schenk-hoppe@leeds.ac.uk
DOI:
https://doi.org/10.1090/S0002-9939-2010-10647-0
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.