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

Fabrizio Lillo, Giulia Livieri, Stefano Marmi, Anton Solomko and Sandro Vaienti

LSE Research Online Documents on Economics from London School of Economics and Political Science, LSE Library

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.

Keywords: leverage cycles; Lyapunov exponents; random dynamical systems; systemic risk; unimodal maps (search for similar items in EconPapers)
JEL-codes: C1 F3 G3 (search for similar items in EconPapers)
Pages: 34 pages
Date: 2023-10-04
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Published in Journal of Statistical Physics, 4, October, 2023, 190(10). ISSN: 0022-4715

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Persistent link: https://EconPapers.repec.org/RePEc:ehl:lserod:120290

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