Skip to Main content Skip to Navigation
Preprints, Working Papers, ...

Nonlinear Randomized Urn Models: a Stochastic Approximation Viewpoint

Abstract : This paper extends the link between stochastic approximation (SA) theory and randomized urn models developed in Laruelle, Pagès (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.
Document type :
Preprints, Working Papers, ...
Complete list of metadatas

https://hal.archives-ouvertes.fr/hal-00910902
Contributor : Sophie Laruelle <>
Submitted on : Friday, May 11, 2018 - 10:43:31 AM
Last modification on : Friday, March 27, 2020 - 3:54:41 AM
Document(s) archivé(s) le : Tuesday, September 25, 2018 - 10:10:32 AM

Files

NonLInearUrns3.pdf
Files produced by the author(s)

Identifiers

  • HAL Id : hal-00910902, version 3
  • ARXIV : 1311.7367

Citation

Sophie Laruelle, Gilles Pagès. Nonlinear Randomized Urn Models: a Stochastic Approximation Viewpoint. 2018. ⟨hal-00910902v3⟩

Share

Metrics

Record views

256

Files downloads

872