Sophie Laruelle

Urn Models and Stochastic Approximation : From Pólya Urns to Nonlinear Randomized Urns

This talk will begin with the presentation of the classic model of Pólya urns and the results obtained to introduce the notations used subsequently. We will then look at variants where we modify the deterministic replacement matrix (without replacement, adding a different color or several colors).
We will then study the case of stochastic replacement matrix with uniform drawing and balanced urn. This has been motivated by applications in clinical trials and finance, and can be solved more efficiently using stochastic algorithms on the dynamics of the proportions of balls in the urn.
We will end this talk with a final generalisation: the drawing rule will no longer be uniform, but will be modified by a convex or concave function. With 2 colors, this will allow us to exhibit a phase transition (in the convex case) passing from one equilibrium to several (the number of which depends on the function). For the number of balls greater than 2, work is in progress, but we have some new results for 3 colors and some ideas we will discuss for going further.