A Weak Bifurcation Theory for Discrete Time Stochastic Dynamical Systems

Diks, Cees G. H.; Wagener, Florian

doi:10.2139/ssrn.901422

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A Weak Bifurcation Theory for Discrete Time Stochastic Dynamical Systems

Tinbergen Institute Discussion Paper No. 06-043/1

33 Pages Posted: 13 May 2006

See all articles by Cees G. H. Diks

Cees G. H. Diks

University of Amsterdam - Faculty of Economics and Business (FEB); Tinbergen Institute

Florian Wagener

University of Amsterdam - Center for Nonlinear Dynamics in Economics and Finance (CeNDEF) - Department of Quantitative Economics; Tinbergen Institute

Date Written: May 2006

Abstract

This article presents a bifurcation theory of smooth stochastic dynamical systems that are governed by everywhere positive transition densities. The local dependence structure of the unique strictly stationary evolution of such a system can be expressed by the ratio of joint and marginal probability densities; this 'dependence ratio' is a geometric invariant of the system. By introducing a weak equivalence notion of these dependence ratios, we arrive at a bifurcation theory for which in the compact case, the set of stable (non-bifurcating) systems is open and dense. The theory is illustrated with some simple examples.

Keywords: Stochastic bifurcation theory

JEL Classification: C14, C22, C32

Suggested Citation: Suggested Citation

Diks, Cees G. H. and Wagener, Florian O.O., A Weak Bifurcation Theory for Discrete Time Stochastic Dynamical Systems (May 2006). Tinbergen Institute Discussion Paper No. 06-043/1, Available at SSRN: https://ssrn.com/abstract=901422 or http://dx.doi.org/10.2139/ssrn.901422