Nonlinear Randomized Urn Models: a Stochastic Approximation Viewpoint
Salle de séminaires M.0.1
This work extends the link between stochastic approximation (SA) theory and randomized urn models, and their applications to clinical trials. We no longer assume that the drawing rule is uniform among the balls of the urn (which contains d colors), but can be reinforced by a function f which models risk aversion. Firstly, by considering that f is concave or convex and by reformulating the dynamics of the urn composition as an SA algorithm with emainder, we derive the a.s. convergence and the asymptotic normality (Central Limit Theorem, CLT) of the normalized procedure by calling upon the so-called ODE and SDE methods. An in-depth analysis of the case d = 2 exhibits two different behaviors: a single equilibrium point when f is concave, and when f is convex, a transition phase from a single attracting equilibrium to a system with two attracting and one repulsive equilibria. The last setting is solved using results on non-convergence toward noisy and noiseless “traps” in order to deduce the a.s. convergence toward one of the attracting points. Finally, these results are applied to an optimal asset allocation in Finance.