On the time and direction of stochastic bifurcation

Abstract

This paper is a mathematical companion to an article introducing a new economics model, by Burdzy, Frankel and Pauzner (1997). The motivation of this paper is applied, but the results may have some mathematical interest in their own right. Our model, i.e., equation (1.1) below, does not seem to be known in literature. Despite its simplicity, it generated some interesting and non-trivial mathematical questions.

In this paper, we limit ourselves to mathematical results; those interested in their economic motivation should consult Burdzy, Frankel and Pauzner (1997). To make this easier, the two papers have been written using comparable notation. A related paper by Bass and Burdzy (1997) will analyze a simplified version of our model and derive a number of new results of a purely mathematical nature.

We will first prove existence and uniqueness for differential equations of the form (1.1) below. These equations involve Brownian motion but they do not fall into the category of classical "stochastic differential equations" as they do not involve the Ito theory of integration. Typical solutions of these equations are Lipschitz functions rather than semi-martingales. It turns out that the excursion theory for Markov processes is the appropriate probabilistic tool for treatment of this family of equations.

We also establish several properties of the "bifurcation time," to be defined below. We prove that the bifurcation time for (1.1) goes to 0 as the random noise becomes smaller and smaller. More importantly, we determine the asymptotic values for probabilities of upward and downward bifurcations.