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Stability analysis with applications of a two-dimensional dynamical system arising from a stochastic model of an asset market

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  • Vladimir Belitsky
  • Antonio L. Pereira
  • Fernando P. de Almeida Prado

Abstract

We analyze the stability properties of equilibrium solutions and periodicity of orbits in a two-dimensional dynamical system whose orbits mimic the evolution of the price of an asset and the excess demand for that asset. The construction of the system is grounded upon a heterogeneous interacting agent model for a single risky asset market. An advantage of this construction procedure is that the resulting dynamical system becomes a macroscopic market model which mirrors the market quantities and qualities that would typically be taken into account solely at the microscopic level of modeling. The system's parameters correspond to: (a) the proportion of speculators in a market; (b) the traders' speculative trend; (c) the degree of heterogeneity of idiosyncratic evaluations of the market agents with respect to the asset's fundamental value; and (d) the strength of the feedback of the population excess demand on the asset price update increment. This correspondence allows us to employ our results in order to infer plausible causes for the emergence of price and demand fluctuations in a real asset market. The employment of dynamical systems for studying evolution of stochastic models of socio-economic phenomena is quite usual in the area of heterogeneous interacting agent models. However, in the vast majority of the cases present in the literature, these dynamical systems are one-dimensional. Our work is among the few in the area that construct and study two-dimensional dynamical systems and apply them for explanation of socio-economic phenomena.

Suggested Citation

  • Vladimir Belitsky & Antonio L. Pereira & Fernando P. de Almeida Prado, 2009. "Stability analysis with applications of a two-dimensional dynamical system arising from a stochastic model of an asset market," Papers 0909.4815, arXiv.org.
  • Handle: RePEc:arx:papers:0909.4815
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    References listed on IDEAS

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    1. Lux, Thomas, 1998. "The socio-economic dynamics of speculative markets: interacting agents, chaos, and the fat tails of return distributions," Journal of Economic Behavior & Organization, Elsevier, vol. 33(2), pages 143-165, January.
    2. Jean-Pierre Nadal & Denis Phan & Mirta Gordon & Jean Vannimenus, 2005. "Multiple equilibria in a monopoly market with heterogeneous agents and externalities," Quantitative Finance, Taylor & Francis Journals, vol. 5(6), pages 557-568.
    3. Alan Kirman, 1993. "Ants, Rationality, and Recruitment," The Quarterly Journal of Economics, Oxford University Press, vol. 108(1), pages 137-156.
    4. Edward L. Glaeser & Jose Scheinkman, 2000. "Non-Market Interactions," NBER Working Papers 8053, National Bureau of Economic Research, Inc.
    5. Gordon, Mirta B. & Nadal, Jean-Pierre & Phan, Denis & Vannimenus, Jean, 2005. "Seller's dilemma due to social interactions between customers," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 356(2), pages 628-640.
    6. Levy, Moshe, 2005. "Social phase transitions," Journal of Economic Behavior & Organization, Elsevier, vol. 57(1), pages 71-87, May.
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