Fokker-Planck equations of jumping particles and mean field games of impulse control

TL;DR

This paper studies the evolution of particle densities under impulse control policies, providing a variational framework for both fixed and density-dependent (mean field) cases, with implications for understanding complex stochastic systems.

Contribution

It introduces a variational characterization of particle densities in impulse control problems, including the mean field game setting, advancing the theoretical understanding of such stochastic processes.

Findings

Derived variational formulations for particle densities

Characterized densities in fixed jump set scenarios

Extended analysis to mean field game of impulse control

Abstract

This paper is interested in the description of the density of particles evolving according to some optimal policy of an impulse control problem. We first fix sets on which the particles jump and explain how we can characterize such a density. We then investigate the coupled case in which the underlying impulse control problem depends on the density we are looking for : the mean field games of impulse control. In both cases, we give a variational characterization of the densities of jumping particles.