Abstract
This paper extends the link between stochastic approximation (SA) theory and randomized urn models developed in Laruelle, Pages (2013), and their applications to clinical trials introduced in Bai, HU (1999,2005) and Bai, Hu, Shen (2002). 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. This is a way to model 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 remainder, 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 equilibrium points. 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. Secondly, the special case of a Polya urn (when the addition rule is the identity matrix) is analyzed, still using result from SA theory about ``traps''. Finally, these results are applied to a function with regular variation and to an optimal asset allocation in Finance.
Highlights
We introduce and study in depth a class of generalized Pólya urns characterized by their nonlinear drawing rules
E-mail: gilles.pages@sorbonne-universite.fr generalization of randomized urn models originally devised for clinical trials which take into account the risk aversion attitude of the prescriber of the treatment
Randomized urn models have been extensively investigated by various authors during the last twenty years based on ad hoc martingale arguments to solve a.s. convergence as well as its rate of convergence
Summary
We introduce and study in depth a class of generalized Pólya urns (with d colors) characterized by their nonlinear drawing rules. Considering nonlinear drawing rules leads, once the evolution of the urn composition is written as a recursive stochastic algorithm of the form (1.1), to consider situations where the mean field function h of the procedure has several zeros, called equilibrium points in SA theory. The interpretation of the skewing functions in terms of risk aversion is the following in the first model: when f is concave, which corresponds to a superlinear reinforcement, the resulting strategy is risk averse and we will see that it tends to equalize the asymptotic urn composition whatever the (irreducible) generating matrix H is. As a consequence, it has a diversifying effect.
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