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Stability Analysis for Random Dynamical Systems in Economics

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Rabi N. Bhattacharya

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Abstract

Random dynamical systems encountered in economics have certain distinctive characteristics that make them particularly well suited to analysis using the tools for studying Markov processes developed by Rabi N. Bhattacharya and his coauthors over the last few decades. In this essay we discuss the significance of these tools for both mathematicians and economists, provide some historical perspective, and review some recent related contributions.

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Notes

  1. 1.

    We focus on discrete time systems since they are more commonly observed in models of economic dynamics. Analysis of continuous time systems can be found, for example, in [13, 2].

  2. 2.

    Subsequently the stability analysis was extended to the case of unbounded shocks by [41, 37, 28, 47].

  3. 3.

    The work of Bhattacharya and Lee builds to some degree on earlier work by Dubins and Freedman [22] and Yahav [46].

  4. 4.

    That is, a ≤ x ≤ b for all x ∈ S.

  5. 5.

    Note that \(\mathbb{P}\{\exists t \geq 0\ \mathrm{s.t.}\ X_{t}'\preceq X_{t}\} = 1\) must also hold by interchanging the two processes.

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Acknowledgements

This paper was written while the second author was visiting RIEB at Kobe University as a Visiting Researcher. Our research has benefited from financial support from the Japan Society for the Promotion of Science and Australian Research Council Discovery Grant DP120100321.

Author information

Authors and Affiliations

  1. Kobe University, Kobe, Japan

    Takashi Kamihigashi

  2. Australian National University, Canberra ACT, Australia

    John Stachurski

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  1. Takashi Kamihigashi
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  2. John Stachurski
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Correspondence to Takashi Kamihigashi .

Editor information

Editors and Affiliations

  1. Mathematics, The Pennsylvania State University, State College, Pennsylvania, USA

    Manfred Denker

  2. Department of Mathematics, Oregon State University, Corvallis, Oregon, USA

    Edward C. Waymire

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Kamihigashi, T., Stachurski, J. (2016). Stability Analysis for Random Dynamical Systems in Economics. In: Denker, M., Waymire, E. (eds) Rabi N. Bhattacharya. Contemporary Mathematicians. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-30190-7_11

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