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 & Continuous Dynamical Systems - A, 2011, 30 (1) : 17-37. doi: 10.3934/dcds.2011.30.17
References:
[1]

L. Arnold, "Random Dynamical Systems,", Springer Verlag, (1998).   Google Scholar

[2]

A. Boyarsky and P. Góra, "Laws of Chaos,", Brikhaüser, (1997).   Google Scholar

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

P. Hall and C. Heyde, "Martingale Limit Theory and Its Application,", Academic Press, (1980).   Google Scholar

[9]

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

[10]

Y. Kifer, "Ergodic Theory of Random Transformations,", Birkh\, (1986).   Google Scholar

[11]

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

[12]

A. N. Shiryaev, "Probability,", Springer-Verlag, (1984).   Google Scholar

[13]

Ö. Stenflo, Uniqueness of invariant measures for place-dependent random iterations of functions,, in, (2002), 13.   Google Scholar

[14]

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

show all references

References:
[1]

L. Arnold, "Random Dynamical Systems,", Springer Verlag, (1998).   Google Scholar

[2]

A. Boyarsky and P. Góra, "Laws of Chaos,", Brikhaüser, (1997).   Google Scholar

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

P. Hall and C. Heyde, "Martingale Limit Theory and Its Application,", Academic Press, (1980).   Google Scholar

[9]

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

[10]

Y. Kifer, "Ergodic Theory of Random Transformations,", Birkh\, (1986).   Google Scholar

[11]

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

[12]

A. N. Shiryaev, "Probability,", Springer-Verlag, (1984).   Google Scholar

[13]

Ö. Stenflo, Uniqueness of invariant measures for place-dependent random iterations of functions,, in, (2002), 13.   Google Scholar

[14]

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

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