June 1st, 2023

18th SICC International Tutorial Workshop "Topics in nonlinear dynamics"
Smooth and Nonsmooth Maps:
Theory, Numerical Methods, and Applications to Economics and Finance

Milan, Università Cattolica del Sacro Cuore, Aula C018, Via Carducci 30

June 1, 2023


Thursday, June 1 (CET time)

8:15-8:45 Registration of the participants
8:45-9:00 Opening
9:00-11:00 Fabio Tramontana
Theory of Nonlinear Smooth Maps
11:00-11:30 Coffee break
11:30-13:00 Iryna Sushko
Nonsmooth maps, continuous and discontinuous: theory and applications
13:00-14:00 Launch break
14:00-15:30 Social event
15:30-16:30 Roberto Dieci
Nonlinear maps in economics and finance: supply, demand, expectations, and price dynamics
16:30-17:00 Coffee break
17:00-18:00 Giulia Livieri
Heteroscedastic noise: Applications to financial systems
18:00 Closing


Abstracts of the contributions

Fabio Tramontana

Università di Urbino, Italy

Theory of Nonlinear Smooth Maps

Nonlinear smooth maps in discrete time naturally arise in social modelling and other fields where the values of the variable cannot change continuously and the state of the system changes at discrete steps. The variables’ motion is driven by systems of difference equations where the next step value of each variable is obtained by using the previous steps values of the same and/or the other variables. In particular, nonlinear smooth maps permit to obtain periodic motion and even chaotic motion already in one dimension, differently from what happens with dynamical systems formulated via differential equations. We will explore the main bifurcation routes to complex dynamics with a particular focus on the coexistence among different attractors and the shapes of their basins of attraction, both in one and two dimensions.

Iryna Sushko

Institute of Mathematics National Academy of Sciences of Ukraine

Nonsmooth maps, continuous and discontinuous: theory and applications

Dynamical systems defined by piecewise smooth maps are nowadays actively studied by many researchers from both theoretical and applied fields. Indeed, since nonlinear dynamics theory has been mainly developed for smooth systems, many theoretical problems associated with nonsmooth functions are still open. Clearly, the nonsmoothness and, especially, the discontinuity of a system enriches its dynamics, leading to new interesting properties. On the other hand, there is a demand for a theory of piecewise smooth dynamical systems from the applied sciences because many important real world processes, characterized by a sharp switching between the states, cannot be represented adequately without the use of piecewise smooth functions. Such functions appear, for example, in modeling dynamics associated with electronic circuits with various kinds of switches, mechanical systems with impacts and/or dry friction, economic and financial processes with different constraints, etc. This lecture aims to present several analytical and numerical tools for studying the dynamics of one- and two-dimensional piecewise smooth maps, continuous and discontinuous, and to give examples of applied models from economics and finance where these tools are successfully used to describe and explain various observed phenomena.

Roberto Dieci

Università di Bologna, Italy

Nonlinear maps in economics and finance: supply, demand, expectations, and price dynamics

Nonlinear smooth systems in discrete time offer the ideal framework to model the dynamic interplay between expectations, demand, supply and market prices in a variety of economic and financial settings characterized by agents' behavioral heterogeneity. Additionally, systems driven by nonsmooth maps can account for the existence of constraints or exogenous policy rules that may induce sharp changes in agents' behavior. This lecture aims to illustrate the effect of such nonlinear interactions in a number of models covering different well-known topics in nonlinear economic dynamics: a stock-flow model of house prices with heterogeneous investors, a cobweb model with naive price expectations and heterogeneous (technologies and) production times, a behavioral asset pricing model with extrapolative expectations and capital gains tax, in which sudden regime changes may arise from a piecewise-smooth map.

Giulia Livieri

London School of Economics, UK

Heteroscedastic noise: Applications to financial systems

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.