Pre-publicaciones Del Seminario Matematico 2002 Urn Models and Differential Algebraic Equations Urn Models and Differential Algebraic Equations *

A generalised urn model is presented in this paper. The urn contains L different types of balls and its replacement policy depends on both an urn function and a random environment. We consider the Ldimensional stochastic process {Xn} that represents the proportion of balls of each type in the urn after each replacement. This process can be expressed as a stochastic recurrent equation that fits a RobbinsMonro scheme. Since the process evolves in the (L − 1)-simplex, the stability of the solutions of the ordinary differential equation associated to the Robbins-Monro scheme can be studied by means of differential algebraic equations (DAE). This approach provides a method to obtain strong laws for the process {Xn} that can be applied even when the number of actions is greater than the number of types of balls and when the total amount of balls added in each step is random. keywords Urn models; Robbins-Monro algorithm; EDO method; differential algebraic equations