How Does Noise Induce Order?

Nisoli, Isaia

doi:10.1007/s10955-022-03041-y

Cited by 5 publications

(1 citation statement)

References 20 publications

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“…Remark 4.8. The Lyapunov exponent (29) was called the average Lyapunov exponent in [38,59]. It was associated with the phenomenon of noise-induced order, which happens when the perturbed systems admit a unique stationary measure depending on some parameter, say θ, and the Lyapunov exponent depends continuously on θ and exhibits a transition from positive to negative values, see also [54] for experimental evidence of this fact.…”

Section: Average Lyapunov Exponentmentioning

confidence: 99%

Analysis of Bank Leverage via Dynamical Systems and Deep Neural Networks

Lillo

Livieri

Marmi

et al. 2023

SIAM J. Finan. Math.

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We consider a model of a simple financial system consisting of a leveraged investor that invests in a risky asset and manages risk by using Value-at-Risk (VaR). The VaR is estimated by using past data via an adaptive expectation scheme. We show that the leverage dynamics can be described by a dynamical system of slow-fast type associated with a unimodal map on [0, 1] with an additive heteroscedastic noise whose variance is related to the portfolio rebalancing frequency to target leverage. In absence of noise the model is purely deterministic and the parameter space splits in two regions: (i) a region with a globally attracting fixed point or a 2-cycle; (ii) a dynamical core region, where the map could exhibit chaotic behavior. Whenever the model is randomly perturbed, we prove the existence of a unique stationary density with bounded variation, the stochastic stability of the process and the almost certain existence and continuity of the Lyapunov exponent for the stationary measure. We then use deep neural networks to estimate map parameters from a short time series. Using this method, we estimate the model in a large dataset of US commercial banks over the period 2001-2014. We find that the parameters of a substantial fraction of banks lie in the dynamical core, and their leverage time series are consistent with a chaotic behavior. We also present evidence that the time series of the leverage of large banks tend to exhibit chaoticity more frequently than those of small banks.

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Section: Average Lyapunov Exponentmentioning

confidence: 99%

Analysis of Bank Leverage via Dynamical Systems and Deep Neural Networks

Lillo

Livieri

Marmi

et al. 2023

SIAM J. Finan. Math.

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Positive Lyapunov Exponent in the Hopf Normal Form with Additive Noise

Chemnitz

Engel

2023

Commun. Math. Phys.

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We prove the positivity of Lyapunov exponents for the normal form of a Hopf bifurcation, perturbed by additive white noise, under sufficiently strong shear strength. This completes a series of related results for simplified situations which we can exploit by studying suitable limits of the shear and noise parameters. The crucial technical ingredient for making this approach rigorous is a result on the continuity of Lyapunov exponents via Furstenberg–Khasminskii formulas.

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亚音速流中二维壁板系统的全局动力学和噪声诱导的跃迁

Yue,

Zhang,

et al. 2024

Acta Mech. Sin.

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How Does Noise Induce Order?

Cited by 5 publications

References 20 publications

Analysis of Bank Leverage via Dynamical Systems and Deep Neural Networks

Analysis of Bank Leverage via Dynamical Systems and Deep Neural Networks

Positive Lyapunov Exponent in the Hopf Normal Form with Additive Noise

亚音速流中二维壁板系统的全局动力学和噪声诱导的跃迁

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