Stochastic ranking process with time dependent intensities

TL;DR

This paper analyzes a stochastic ranking process with particles whose jump times are governed by Poisson measures, establishing convergence of the empirical distribution and deriving an explicit limit distribution via PDEs.

Contribution

It introduces a new model with time-dependent intensities and provides a rigorous proof of convergence and explicit characterization of the limit distribution.

Findings

Empirical distribution converges almost surely as particle number grows.

Explicit formula for the limit distribution is derived.

Limit distribution solves a system of nonlinear PDEs with time-dependent coefficients.

Abstract

We consider the stochastic ranking process with the jump times of the particles determined by Poisson random measures. We prove that the joint empirical distribution of scaled position and intensity measure converges almost surely in the infinite particle limit. We give an explicit formula for the limit distribution and show that the limit distribution function is a unique global classical solution to an initial value problem for a system of a first order non-linear partial differential equations with time dependent coefficients.