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PRIVILEGGI Fabio and Luigi MONTRUCCHIO (1999), "Fractal Steady States in Stochastic Optimal Control Models", Annals of Operations Research, 88 (0), 183-197.
Keywords: stochastic dynamic programming, chaotic dynamics, fractals, invariant probabilities

The paper is divided into two parts. We first extend the Boldrin and Montrucchio theorem [5] on the inverse control problem to the Markovian stochastic setting. Given a dynamical system xt+1 = g(xt, zt), we find a discount factor b* such that for each 0 < b < b* 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 selfsimilarities. A neo-classical three-sector stochastic optimal growth exhibiting the Sierpinski carpet as the unique attractor is provided as an example.

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