[Paper Review] Fokker-Planck equations of jumping particles and mean field games of impulse control.

Charles Bertucci|ArXiv.org|Mar 14, 2018

Stochastic processes and financial applicationsEconomics, Econometrics and Finance8 references4 citations

TL;DR

This paper establishes a variational characterization of the density of particles in impulse control systems, deriving a Fokker-Planck-type equation for jumping particles and formulating mean field games (MFG) of impulse control via quasi-variational inequalities. The key contribution is proving that the unique solution to the MFG system corresponds to the unique minimizer of a convex functional, providing a variational framework for equilibrium analysis in MFG with jumps.

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.

Motivation & Objective

• To characterize the density of particles evolving under impulse control in a stochastic diffusion process with jumps.
• To develop a Fokker-Planck equation framework for systems where particles jump at controlled times and locations.
• To extend mean field game theory to impulse control settings, where individual player costs depend on the population density.
• To establish existence and uniqueness of solutions for the coupled forward-backward MFG system in the impulse control context.
• To provide a variational formulation of the equilibrium density as the unique minimizer of a convex functional involving value functions and density constraints.

Proposed method

• Formulates a penalized version of the impulse control problem to derive a limit density in the time-dependent case.
• Uses a variational approach to characterize the equilibrium density as the minimizer of a functional involving the value function and density constraints.
• Applies weak convergence and lower/upper semi-continuity arguments in Sobolev spaces to prove existence and uniqueness of the limit density.
• Derives a system of variational inequalities that characterize the solution of the MFG system, linking the value function and density evolution.
• Establishes a saddle-point formulation of the MFG system through a convex-concave functional, ensuring uniqueness via strict convexity.
• Extends the framework to the stationary case, proving existence and uniqueness of the stationary Fokker-Planck equation under appropriate conditions.

Experimental results

Research questions

• RQ1 How can the density of particles in a system of jumping particles governed by impulse control be mathematically characterized?
• RQ2 What is the structure of the Fokker-Planck equation when the particle dynamics involve controlled jumps rather than continuous diffusion?
• RQ3 How can mean field games with impulse control be formulated as a coupled system of forward-backward equations?
• RQ4 What variational principles underlie the equilibrium density in mean field games of impulse control?
• RQ5 Under what conditions does a unique equilibrium density exist in such systems?

Key findings

• The limit density of jumping particles is characterized as the unique minimizer of a convex functional involving the value function and density constraints.
• The solution to the mean field game system of impulse control is shown to be the unique minimizer of a variational problem, establishing a variational characterization of equilibrium.
• Existence and uniqueness of the solution to the stationary Fokker-Planck equation for impulse control are proven under appropriate assumptions.
• The system of variational inequalities derived from the MFG formulation characterizes the equilibrium as a saddle point of a convex-concave functional.
• The penalized problem converges to a solution that satisfies the Fokker-Planck equation for jumping particles, with convergence established via weak compactness and semi-continuity.
• The optimal control interpretation of the MFG system is shown to correspond to minimizing a functional involving the running cost and a dual term related to the density constraint.