A Discrete--Delay Dynamic Model for the Stock Market
The time evolution of prices and savings in a stock market is modeled by a discrete-delay nonlinear dynamic system. The proposed model has a unique and unstable steady-state, so its time evolution is determined by the nonlinear effects acting out of the equilibrium. We perform the analysis of the linear approximation through the study of the eigenvalues of the Jacobian matrix in order to characterize the local stability properties and the local bifurcations in the parameter space. If the delay is equal to zero, Lyapunov exponents are calculated. For certain values of the parameters, we prove that the system has a chaotic behaviour. The discrete nonlinear model is associated with a discrete stochastic model. For the liniarization of this model, we establish the conditions for which the mean and quadratic mean values of the state variables are asymptotically stable. Some numerical examples are finally given to justify the theoretical results.
|Date of creation:||Oct 2011|
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- Roberto Dieci & Ilaria Foroni & Laura Gardini & Xue-Zhong He, 2005. "Market Mood, Adaptive Beliefs and Asset Price Dynamics," Research Paper Series 162, Quantitative Finance Research Centre, University of Technology, Sydney.
- Gian-Italo Bischi & Vincenzo Valori, 2000. "Nonlinear effects in a discrete-time dynamic model of a stock market," Working Papers - Mathematical Economics 2000-01, Universita' degli Studi di Firenze, Dipartimento di Scienze per l'Economia e l'Impresa.
- Carl Chiarella & Xue-Zhong He & Duo Wang, 2004. "A Behavioural Asset Pricing Model with a Time-Varying Second Moment," Research Paper Series 141, Quantitative Finance Research Centre, University of Technology, Sydney.
- Babus, Ana & de Vries, Casper G., 2010. "Global stochastic properties of dynamic models and their linear approximations," Journal of Economic Dynamics and Control, Elsevier, vol. 34(5), pages 817-824, May.
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