April  2011, 30(1): 17-37. doi: 10.3934/dcds.2011.30.17

SRB measures for certain Markov processes

1. 

Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire, LE11 3TU

2. 

Department of Mathematics and Statistics, Concordia University, Montreal, Quebec, H3G 1M8, Canada

Received  January 2010 Revised  July 2010 Published  February 2011

We study Markov processes generated by iterated function systems (IFS). The constituent maps of the IFS are monotonic transformations of the interval. We first obtain an upper bound on the number of SRB (Sinai-Ruelle-Bowen) measures for the IFS. Then, when all the constituent maps have common fixed points at 0 and 1, theorems are given to analyze properties of the ergodic invariant measures $\delta_0$ and $\delta_1$. In particular, sufficient conditions for $\delta_0$ and/or $\delta_1$ to be, or not to be, SRB measures are given. We apply some of our results to asset market games.
Citation: Wael Bahsoun, Paweł Góra. SRB measures for certain Markov processes. Discrete and Continuous Dynamical Systems, 2011, 30 (1) : 17-37. doi: 10.3934/dcds.2011.30.17
References:
[1]

L. Arnold, "Random Dynamical Systems," Springer Verlag, Berlin, 1998.

[2]

A. Boyarsky and P. Góra, "Laws of Chaos," Brikhäuser, Boston, 1997.

[3]

J. Buzzi, Absolutely continuous S.R.B. measures for random Lasota-Yorke maps, Trans. Amer. Math. Soc., 352 (2000), 3289-3303. doi: 10.1090/S0002-9947-00-02607-6.

[4]

I. Evstigneev, T. Hens and K. R. Schenk-Hoppé, Market selection of financial trading strategies: Global stability, Math. Finance, 12 (2002), 329-339. doi: 10.1111/j.1467-9965.2002.tb00127.x.

[5]

P. Diaconis and D. Freedman, Iterated random functions, SIAM Rev., 41 (1999), 45-76. doi: 10.1137/S0036144598338446.

[6]

R. Drogin, An invariance principle for martingales, Ann. Math. Statist., 43 (1972), 602-620. doi: 10.1214/aoms/1177692640.

[7]

L. Dubins and D. Freedman, Invariant probabilities for certain Markov processes, Ann. Math. Statist., 37 (1966), 837-848. doi: 10.1214/aoms/1177699364.

[8]

P. Hall and C. Heyde, "Martingale Limit Theory and Its Application," Academic Press, New York-London, 1980.

[9]

J. L. Kelly, A new interpretation of information rate, Bell Sys. Tech. J., 35 (1956), 917-926.

[10]

Y. Kifer, "Ergodic Theory of Random Transformations," Birkhäuser, Boston, 1986.

[11]

P.-D. Liu, Dynamics of random transformations: smooth ergodic theory, Ergodic Theory Dynam. Syst., 21 (2001), 1279-1319. doi: 10.1017/S0143385701001614.

[12]

A. N. Shiryaev, "Probability," Springer-Verlag, New York, 1984.

[13]

Ö. Stenflo, Uniqueness of invariant measures for place-dependent random iterations of functions, in "Fractals in Multimedia" (eds. M. F. Barnsley, D. Saupe and E. R. Vrscay), Springer, (2002), 13-32.

[14]

L.-S. Young, What are SRB measures, and which dynamical systems have them?, J. Statist. Phys., 108 (2002), 733-754. doi: 10.1023/A:1019762724717.

show all references

References:
[1]

L. Arnold, "Random Dynamical Systems," Springer Verlag, Berlin, 1998.

[2]

A. Boyarsky and P. Góra, "Laws of Chaos," Brikhäuser, Boston, 1997.

[3]

J. Buzzi, Absolutely continuous S.R.B. measures for random Lasota-Yorke maps, Trans. Amer. Math. Soc., 352 (2000), 3289-3303. doi: 10.1090/S0002-9947-00-02607-6.

[4]

I. Evstigneev, T. Hens and K. R. Schenk-Hoppé, Market selection of financial trading strategies: Global stability, Math. Finance, 12 (2002), 329-339. doi: 10.1111/j.1467-9965.2002.tb00127.x.

[5]

P. Diaconis and D. Freedman, Iterated random functions, SIAM Rev., 41 (1999), 45-76. doi: 10.1137/S0036144598338446.

[6]

R. Drogin, An invariance principle for martingales, Ann. Math. Statist., 43 (1972), 602-620. doi: 10.1214/aoms/1177692640.

[7]

L. Dubins and D. Freedman, Invariant probabilities for certain Markov processes, Ann. Math. Statist., 37 (1966), 837-848. doi: 10.1214/aoms/1177699364.

[8]

P. Hall and C. Heyde, "Martingale Limit Theory and Its Application," Academic Press, New York-London, 1980.

[9]

J. L. Kelly, A new interpretation of information rate, Bell Sys. Tech. J., 35 (1956), 917-926.

[10]

Y. Kifer, "Ergodic Theory of Random Transformations," Birkhäuser, Boston, 1986.

[11]

P.-D. Liu, Dynamics of random transformations: smooth ergodic theory, Ergodic Theory Dynam. Syst., 21 (2001), 1279-1319. doi: 10.1017/S0143385701001614.

[12]

A. N. Shiryaev, "Probability," Springer-Verlag, New York, 1984.

[13]

Ö. Stenflo, Uniqueness of invariant measures for place-dependent random iterations of functions, in "Fractals in Multimedia" (eds. M. F. Barnsley, D. Saupe and E. R. Vrscay), Springer, (2002), 13-32.

[14]

L.-S. Young, What are SRB measures, and which dynamical systems have them?, J. Statist. Phys., 108 (2002), 733-754. doi: 10.1023/A:1019762724717.

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