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We introduce and study ergodic multidimensional diffusion processes interacting through their ranks; these interactions lead to invariant measures which are in broad agreement with stability properties of large equity markets over long time-periods.
The models we develop assign growth rates and variances that depend on both the name (identity) and the rank (according to capitalization) of each individual asset.
Such models are able realistically to capture critical features of the observed stability of capital distribution over the past century, all the while being simple enough to allow for rather detailed analytical study.
The methodologies used in this study touch upon the question of triple points for systems of interacting diffusions; in particular, some choices of parameters may permit triple (or higher-order) collisions to occur. We show, however, that such multiple collisions have no effect on any of the stability properties of the resulting system. This is accomplished through a detailed analysis of intersection local times.
The theory we develop has connections with the analysis of Queueing Networks in heavy traffic, as well as with models of competing particle systems in Statistical Mechanics, such as the Sherrington-Kirkpatrick model for spin-glasses.