IDEAS home Printed from https://ideas.repec.org/p/arx/papers/2305.13475.html

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

Author

Listed:
  • F. Lillo
  • G. Livieri
  • S. Marmi
  • A. Solomko
  • S. Vaienti

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-Ess\`een 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

  • F. Lillo & G. Livieri & S. Marmi & A. Solomko & S. Vaienti, 2023. "Unimodal maps perturbed by heteroscedastic noise: an application to a financial systems," Papers 2305.13475, arXiv.org.
    • Handle: RePEc:arx:papers:2305.13475
    as

    Download full text from publisher

    File URL: https://arxiv.org/pdf/2305.13475
    File Function: Latest version
    Download Restriction: no
    ---><---

    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. Jarque, Carlos M. & Bera, Anil K., 1980. "Efficient tests for normality, homoscedasticity and serial independence of regression residuals," Economics Letters, Elsevier, vol. 6(3), pages 255-259.
    3. 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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Lillo, Fabrizio & Livieri, Giulia & 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.
    2. Fabrizio Lillo & Giulia Livieri & Stefano Marmi & Anton Solomko & Sandro Vaienti, 2021. "Analysis of bank leverage via dynamical systems and deep neural networks," Papers 2104.04960, arXiv.org.
    3. 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.
    4. 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.
    5. Bierens, H.J. & Broersma, L., 1991. "The relation between unemployment and interest rate : some international evidence," Serie Research Memoranda 0112, VU University Amsterdam, Faculty of Economics, Business Administration and Econometrics.
    6. Ha, Le Thanh, 2025. "From wars to dynamic waves: Scrutinizing connectedness between geopolitical risk index, green and non-green crypto volatility by quantile spillovers," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 679(C).
    7. Halldén, Filip & Hultberg, Anna & Ahmed, Ali & Uddin, Gazi Salah & Yahya, Muhammad & Troster, Victor, 2025. "The role of institutional quality on public renewable energy investments," Renewable and Sustainable Energy Reviews, Elsevier, vol. 215(C).
    8. Juncal Cunado & David Gabauer & Rangan Gupta, 2024. "Realized volatility spillovers between energy and metal markets: a time-varying connectedness approach," Financial Innovation, Springer;Southwestern University of Finance and Economics, vol. 10(1), pages 1-17, December.
    9. Elie Bouri & Georges Azzi, 2014. "On the Dynamic Transmission of Mean and Volatility across the Arab Stock Markets," Journal of Emerging Market Finance, Institute for Financial Management and Research, vol. 13(3), pages 279-304, December.
    10. Eric Fur, 2023. "Risk and return of classic car market prices: passion or financial investment?," Journal of Asset Management, Palgrave Macmillan, vol. 24(1), pages 59-68, February.
    11. Abdullah, Mohammad & Chowdhury, Mohammad Ashraful Ferdous & Wali Ullah, G.M., 2025. "Asymmetric tail risk dynamics, efficiency and risk spillover among FinTech stocks, cryptocurrencies and traditional assets," Global Finance Journal, Elsevier, vol. 64(C).
    12. Georgiev, Iliyan, 2010. "Model-based asymptotic inference on the effect of infrequent large shocks on cointegrated variables," Journal of Econometrics, Elsevier, vol. 158(1), pages 37-50, September.
    13. Jin, Xiaoye, 2015. "Volatility transmission and volatility impulse response functions among the Greater China stock markets," Journal of Asian Economics, Elsevier, vol. 39(C), pages 43-58.
    14. Andrea Flori & Fabrizio Lillo & Fabio Pammolli & Alessandro Spelta, 2021. "Better to stay apart: asset commonality, bipartite network centrality, and investment strategies," Annals of Operations Research, Springer, vol. 299(1), pages 177-213, April.
    15. G. D. Gettinby & C. D. Sinclair & D. M. Power & R. A. Brown, 2004. "An Analysis of the Distribution of Extreme Share Returns in the UK from 1975 to 2000," Journal of Business Finance & Accounting, Wiley Blackwell, vol. 31(5‐6), pages 607-646, June.
    16. Damian Jelito & Marcin Pitera, 2018. "New fat-tail normality test based on conditional second moments with applications to finance," Papers 1811.05464, arXiv.org, revised Apr 2020.
    17. Ilham Haouas & Naceur Kheraief & Arusha Cooray & Syed Jawad Hussain Shahzad, 2019. "Time-Varying Casual Nexuses Between Remittances and Financial Development in Some MENA Countries," Working Papers 1294, Economic Research Forum, revised 2019.
    18. Stephen Keef & Melvin Roush, 2005. "Day-of-the-week effects in the pre-holiday returns of the Standard & Poor's 500 stock index," Applied Financial Economics, Taylor & Francis Journals, vol. 15(2), pages 107-119.
    19. Yuan, Di & Li, Sufang & Li, Rong & Zhang, Feipeng, 2022. "Economic policy uncertainty, oil and stock markets in BRIC: Evidence from quantiles analysis," Energy Economics, Elsevier, vol. 110(C).
    20. Benedikt Gasser & Joel Stouder, 2022. "Rescue Emergencies Due to High-Altitude Illnesses Are Rare in Switzerland," IJERPH, MDPI, vol. 19(2), pages 1-7, January.

    More about this item

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:arx:papers:2305.13475. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: arXiv administrators (email available below). General contact details of provider: https://arxiv.org/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.