Fractal steady states in stochastic optimal control models

The paper is divided into two parts. We first extend the Boldrin and Montrucchio theorem on the inverse control problem to the Markovian stochastic setting. Given a dynamical system xt+1 = g(xt, zt), we find a discount factor β^* such that for each 0 < β < β^* a concave problem exists for which the dynamical system is an optimal solution. In the second part, we use the previous result for constructing stochastic optimal control systems having fractal attractors. In order to do this, we rely on some results by Hutchinson on fractals and self-similarities. A neo-classical three-sector stochastic optimal growth exhibiting the Sierpinski carpet as the unique attractor is provided as an example.