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 into 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.


  1. leverage cycles
  2. risk management
  3. systemic risk
  4. random dynamical systems
  5. unimodal maps
  6. Lyapunov exponents
  7. neural networks

MSC codes

  1. 91G80
  2. 34F05
  3. 37H15
  4. 62M45

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S.V. thanks C. Gonzalez-Tokman for useful discussion about subsection 3.4.


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Information & Authors


Published In

cover image SIAM Journal on Financial Mathematics
SIAM Journal on Financial Mathematics
Pages: 598 - 643
ISSN (online): 1945-497X


Submitted: 15 April 2021
Accepted: 13 February 2023
Published online: 31 May 2023


  1. leverage cycles
  2. risk management
  3. systemic risk
  4. random dynamical systems
  5. unimodal maps
  6. Lyapunov exponents
  7. neural networks

MSC codes

  1. 91G80
  2. 34F05
  3. 37H15
  4. 62M45



Dipartimento di Matematica, Università di Bologna, Bologna, 40126, and Scuola Normale Superiore, Pisa, 56127, Italy.
The London School of Economics and Political Science, London, United Kingdom.
Stefano Marmi
Scuola Normale Superiore, Pisa, 56127, Italy.
Anton Solomko
Envelop Risk, Bristol, UK.
Sandro Vaienti
Aix Marseille Université, Université de Toulon, CNRS, CPT, 13009 Marseille, France.

Funding Information

Istituto Nazionale di Alta Matematica
Funding: This research was supported by the research project “Dynamics and Information Research Institute - Quantum Information, Quantum Technologies” within the agreement between UniCredit Bank and Scuola Normale Superiore. The first and third authors were partially supported by the European Program scheme “INFRAIA-01-2018-2019: Research and Innovation action,” grant agreement 871042, “SoBigData++: European Integrated Infrastructure for Social Mining and Big Data Analytics.” The fifth author received support from the Centro di Ricerca Matematica Ennio de Giorgi, Scuola Normale Superiore, the Laboratoire International Associé LIA LYSM, INdAM (Italy), and the UMI-CNRS 3483, Laboratoire Fibonacci (Pisa), where this work was initiated and completed under a CNRS delegation.

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