Mathematical Finance Seminar
18 c.t.
TU Berlin, Room MA 313 (Straße des 17. Juni 136, 10623 Berlin)
Wen Sun (University Pierre and Marie CURIE, Paris)

Large urn model with local mean-field interactions

We study a large urn model connected by an underlying symmetrical graph. After some exponentially distributed amount of time all the balls of one of the urns are redistributed among the connected urns. The allocation of balls is done at random according to a set of weights which depend on the state of the system. The degree of the graph, which is the range of interaction, is assumed large, but is not necessarily linear with respect to the number of urns. Moreover, the number of simultaneous jumps of the process is not bounded due to the redistribution of all balls of an urn at the same time. We describe the dynamic by using the local empirical distributions associated to the state of urns in the neighborhood of a given urn. Under some conditions, we are able to establish a mean-field convergence result for this measure-valued system. Convergence results of the corresponding invariant distributions are obtained for several classes of allocation policies. For the  class of power of choices policies, we show that the invariant measure has an asymptotic finite support property when the average load per urn gets  large. This result differs somewhat from the classical double exponential decay property usually encountered in the literature for power of choices policies.  This finite support property has interesting consequences in practice. (This is a joint work with Philippe Robert.)