Abstract
A time‐delayed model of speculative asset markets is investigated to discuss the effect of time delay and market fraction of the fundamentalists on the dynamics of asset prices. It proves that a sequence of Hopf bifurcations occurs at the positive equilibrium v, the fundamental price of the asset, as the parameters vary. The direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are determined using normal form method and center manifold theory. Global existence of periodic solutions is established combining a global Hopf bifurcation theorem with a Bendixson′s criterion for higher‐dimensional ordinary differential equations.
Highlights
Efficient Market Hypothesis EMH is a standard theory of financial market dynamics
Asset prices follow a geometric Brownian motion representing the fundamental value of the asset, and asset prices cannot deviate from their fundamental values
We investigate the dependence of local dynamics of P on parameters m and τ
Summary
Efficient Market Hypothesis EMH is a standard theory of financial market dynamics. According to the theory, asset prices follow a geometric Brownian motion representing the fundamental value of the asset, and asset prices cannot deviate from their fundamental values. Models have been developed to explain fluctuations in financial markets see 1–9 and the references therein In such models, asset prices follow deterministic paths that can deviate from fundamental values generating what is called a speculative bubble in asset markets. Applying the local Hopf bifurcation theory see 10 , we investigate the existence of periodic oscillations for P , which depends both on time delay τ and the market fraction of the fundamentalists m. A key step in establishing the global extension of the local Hopf branch at the first critical value τ τ0 is to verify that P has no nonconstant periodic solutions of period 4τ This is accomplished by applying a higher-dimensional Bendixson’s criterion for ordinary differential equations given by Li and Muldowney.
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