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Time operator of Markov chains and mixing times. Applications to financial data

Author

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  • Gialampoukidis, I.
  • Gustafson, K.
  • Antoniou, I.

Abstract

We extend the notion of Time Operator from Kolmogorov Dynamical Systems and Bernoulli processes to Markov processes. The general methodology is presented and illustrated in the simple case of binary processes. We present a method to compute the eigenfunctions of the Time Operator. Internal Ages are related to other characteristic times of Markov chains, namely the Kemeny time, the convergence rate and Goodman’s intrinsic time. We clarified the concept of mixing time by providing analytic formulas for two-state Markov chains. Explicit formulas for mixing times are presented for any two-state regular Markov chain. The mixing time of a Markov chain is determined also by the Time Operator of the Markov chain, within its Age computation. We illustrate these results in terms of two realistic examples: A Markov chain from US GNP data and a Markov chain from Dow Jones closing prices. We propose moreover a representation for the Kemeny constant, in terms of internal Ages.

Suggested Citation

  • Gialampoukidis, I. & Gustafson, K. & Antoniou, I., 2014. "Time operator of Markov chains and mixing times. Applications to financial data," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 415(C), pages 141-155.
    • Handle: RePEc:eee:phsmap:v:415:y:2014:i:c:p:141-155
    DOI: 10.1016/j.physa.2014.07.084
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    References listed on IDEAS

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    1. Antoniou, I. & Gustafson, K., 1999. "Wavelets and stochastic processes," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 49(1), pages 81-104.
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    8. Gialampoukidis, I. & Gustafson, K. & Antoniou, I., 2013. "Financial Time Operator for random walk markets," Chaos, Solitons & Fractals, Elsevier, vol. 57(C), pages 62-72.
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    Cited by:

    1. Gialampoukidis, Ilias & Antoniou, Ioannis, 2015. "Age, Innovations and Time Operator of Networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 432(C), pages 140-155.

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