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Convergence of locally and globally interacting Markov chains

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  • Föllmer, Hans
  • Horst, Ulrich

Abstract

We study the long run behaviour of interactive Markov chains on infinite product spaces. In view of microstructure models of financial markets, the interaction has both a local and a global component. The convergence of such Markov chains is analyzed on the microscopic level and on the macroscopic level of empirical fields. We give sufficient conditions for convergence on the macroscopic level. Using a perturbation of the Dobrushin-Vasserstein contraction technique we show that macroscopic convergence implies weak convergence of the underlying Markov chain. This extends the basic convergence theorem of Vasserstein (1969) for locally interacting Markov chains to the case where an additional global component appears in the interaction.

Suggested Citation

  • Föllmer, Hans & Horst, Ulrich, 2001. "Convergence of locally and globally interacting Markov chains," SFB 373 Discussion Papers 2001,21, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
    • Handle: RePEc:zbw:sfb373:200121
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    Cited by:

    1. Horst, Ulrich & Scheinkman, Jose A., 2006. "Equilibria in systems of social interactions," Journal of Economic Theory, Elsevier, vol. 130(1), pages 44-77, September.
    2. Biao Wu, 2007. "Interacting Agent Feedback Finance Model," Papers math/0703827, arXiv.org.
    3. ÖZGÜR, Onur & BISIN, Alberto, 2011. "Dynamic Linear Economies with Social Interactions," Cahiers de recherche 04-2011, Centre interuniversitaire de recherche en économie quantitative, CIREQ.
    4. Agudze, Komla M. & Billio, Monica & Casarin, Roberto & Ravazzolo, Francesco, 2022. "Markov switching panel with endogenous synchronization effects," Journal of Econometrics, Elsevier, vol. 230(2), pages 281-298.
    5. Alberto Bisin & Andrea Moro & Giorgio Topa, 2011. "The empirical content of models with multiple equilibria in economies with social interactions," Staff Reports 504, Federal Reserve Bank of New York.
    6. Horst, Ulrich, 2010. "Dynamic systems of social interactions," Journal of Economic Behavior & Organization, Elsevier, vol. 73(2), pages 158-170, February.
    7. Erhan Bayraktar & Ulrich Horst & Ronnie Sircar, 2007. "Queueing Theoretic Approaches to Financial Price Fluctuations," Papers math/0703832, arXiv.org.
    8. Fu, Guanxing & Horst, Ulrich, 2017. "Mean Field Games with Singular Controls," Rationality and Competition Discussion Paper Series 22, CRC TRR 190 Rationality and Competition.
    9. Bisin, Alberto & Horst, Ulrich & Ozgur, Onur, 2006. "Rational expectations equilibria of economies with local interactions," Journal of Economic Theory, Elsevier, vol. 127(1), pages 74-116, March.
    10. Alberto Bisin & Thierry Verdier, 2010. "The Economics of Cultural Transmission and Socialization," Post-Print halshs-00754788, HAL.
    11. Ulrich Horst & Jos´e A. Scheinkman, 2006. "A Limit Theorem for Systems of Social Interactions," Levine's Bibliography 321307000000000177, UCLA Department of Economics.
    12. Horst, Ulrich & Scheinkman, José A., 2009. "A limit theorem for systems of social interactions," Journal of Mathematical Economics, Elsevier, vol. 45(9-10), pages 609-623, September.

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