Large systems of diffusions interacting through their ranks

TL;DR

This paper analyzes the limiting behavior of large systems of diffusions interacting through their ranks, showing they converge to a McKean-Vlasov evolution governed by a generalized porous medium equation, with applications in financial market models.

Contribution

It establishes the convergence of rank-based diffusions to a McKean-Vlasov equation and links the dynamics to a generalized porous medium equation, providing a new theoretical framework.

Findings

Proves convergence of empirical measures to a McKean-Vlasov equation.

Shows the evolution of the distribution follows a generalized porous medium equation.

Establishes uniqueness of solutions and law of large numbers for the system.

Abstract

We study the limiting behaviour of the empirical measure of a system of diffusions interacting through their ranks when the number of diffusions tends to infinity. We prove that the limiting dynamics is given by a McKean-Vlasov evolution equation. Moreover, we show that in a wide range of cases the evolution of the cumulative distribution function under the limiting dynamics is governed by the generalized porous medium equation with convection. The uniqueness theory for the latter is used to establish the uniqueness of solutions of the limiting McKean-Vlasov equation and the law of large numbers for the corresponding systems of interacting diffusions. The implications of the results for rank-based models of capital distributions in financial markets are also explained.