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Unimodal Maps Perturbed by Heteroscedastic Noise: An Application to a Financial Systems

Author

Listed:
  • Fabrizio Lillo

    (UNIBO - Alma Mater Studiorum Università di Bologna = University of Bologna)

  • Giulia Livieri

    (LSE - London School of Economics and Political Science)

  • Stefano Marmi

    (SNS - Scuola Normale Superiore di Pisa)

  • Anton Solomko
  • Sandro Vaienti

    (CPT - Centre de Physique Théorique - UMR 7332 - AMU - Aix Marseille Université - UTLN - Université de Toulon - CNRS - Centre National de la Recherche Scientifique)

Abstract

We investigate and prove the mathematical properties of a general class of one-dimensional unimodal smooth maps perturbed with a heteroscedastic noise. Specifically, we investigate the stability of the associated Markov chain, show the weak convergence of the unique stationary measure to the invariant measure of the map, and show that the average Lyapunov exponent depends continuously on the Markov chain parameters. Representing the Markov chain in terms of random transformation enables us to state and prove the Central Limit Theorem, the large deviation principle, and the Berry-Esséen inequality. We perform a multifractal analysis for the invariant and the stationary measures, and we prove Gumbel's law for the Markov chain with an extreme index equal to 1. In addition, we present an example linked to the financial concept of systemic risk and leverage cycle, and we use the model to investigate the finite sample properties of our asymptotic results

Suggested Citation

  • Fabrizio Lillo & Giulia Livieri & Stefano Marmi & Anton Solomko & Sandro Vaienti, 2023. "Unimodal Maps Perturbed by Heteroscedastic Noise: An Application to a Financial Systems," Post-Print hal-04389232, HAL.
    • Handle: RePEc:hal:journl:hal-04389232
    DOI: 10.1007/s10955-023-03160-0
    Note: View the original document on HAL open archive server: https://hal.science/hal-04389232
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    References listed on IDEAS

    as
    1. Fulvio Corsi & Stefano Marmi & Fabrizio Lillo, 2016. "When Micro Prudence Increases Macro Risk: The Destabilizing Effects of Financial Innovation, Leverage, and Diversification," Operations Research, INFORMS, vol. 64(5), pages 1073-1088, October.
    2. Mazzarisi, Piero & Lillo, Fabrizio & Marmi, Stefano, 2019. "When panic makes you blind: A chaotic route to systemic risk," Journal of Economic Dynamics and Control, Elsevier, vol. 100(C), pages 176-199.
    3. Lillo, Fabrizio & Livieri, Giulia & Marmi, Stefano & Solomko, Anton & Vaienti, Sandro, 2023. "Analysis of bank leverage via dynamical systems and deep neural networks," LSE Research Online Documents on Economics 119917, London School of Economics and Political Science, LSE Library.
    Full references (including those not matched with items on IDEAS)

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