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

Fabrizio Lillo, Giulia Livieri, Stefano Marmi, Anton Solomko and Sandro Vaienti ()
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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
Sandro Vaienti: CPT - Centre de Physique Théorique - UMR 7332 - AMU - Aix Marseille Université - UTLN - Université de Toulon - CNRS - Centre National de la Recherche Scientifique

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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

Keywords: random dynamical systems; unimodal maps; Lyapunov exponents; leverage cycles; systemic risk (search for similar items in EconPapers)
Date: 2023
Note: View the original document on HAL open archive server: https://hal.science/hal-04389232
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Published in Journal of Statistical Physics, 2023, 190 (10), pp.156. ⟨10.1007/s10955-023-03160-0⟩

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Persistent link: https://EconPapers.repec.org/RePEc:hal:journl:hal-04389232

DOI: 10.1007/s10955-023-03160-0

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