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Unimodal maps perturbed by heteroscedastic noise: an application to a financial systems

Author

Listed:
  • Livieri, Giulia
  • Lillo, Fabrizio
  • Marmi, Stefano
  • Solomko, Anton
  • Vaienti, Sandro

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.

Suggested Citation

  • Livieri, Giulia & Lillo, Fabrizio & Marmi, Stefano & Solomko, Anton & Vaienti, Sandro, 2023. "Unimodal maps perturbed by heteroscedastic noise: an application to a financial systems," LSE Research Online Documents on Economics 120290, London School of Economics and Political Science, LSE Library.
    • Handle: RePEc:ehl:lserod:120290
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    File URL: http://eprints.lse.ac.uk/120290/
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    References listed on IDEAS

    as
    1. Freitas, Ana Cristina Moreira & Freitas, Jorge Milhazes, 2008. "On the link between dependence and independence in extreme value theory for dynamical systems," Statistics & Probability Letters, Elsevier, vol. 78(9), pages 1088-1093, July.
    2. 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.
    3. Zhang, Lan & Mykland, Per A. & Ait-Sahalia, Yacine, 2005. "A Tale of Two Time Scales: Determining Integrated Volatility With Noisy High-Frequency Data," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 1394-1411, December.
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    5. 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.
    6. Tobias Adrian & Erkko Etula & Tyler Muir, 2014. "Financial Intermediaries and the Cross-Section of Asset Returns," Journal of Finance, American Finance Association, vol. 69(6), pages 2557-2596, December.
    7. Paul Embrechts & Sidney Resnick & Gennady Samorodnitsky, 1999. "Extreme Value Theory as a Risk Management Tool," North American Actuarial Journal, Taylor & Francis Journals, vol. 3(2), pages 30-41.
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    More about this item

    Keywords

    leverage cycles; Lyapunov exponents; random dynamical systems; systemic risk; unimodal maps; Springer deal;
    All these keywords.

    JEL classification:

    • C1 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General
    • F3 - International Economics - - International Finance
    • G3 - Financial Economics - - Corporate Finance and Governance

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