We study an ordinary differential equation controlled by a stochastic process. We present results on existence and uniqueness of solutions, on associated local times (Trotter and Ray-Knight theorems), and on time and direction of bifurcation. A relationship with Lipschitz approximations to Brownian paths is also discussed.

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 ...