Asymptotics for a free-boundary model in price formation

TL;DR

This paper analyzes the long-term behavior of solutions to a mean-field free boundary model in price formation, showing stability and exponential decay to equilibrium using semigroup and center manifold theories.

Contribution

It provides the first rigorous analysis of the asymptotics and stability of solutions in a non-standard free boundary model related to price formation.

Findings

Solutions decay exponentially to equilibrium.

The free boundary disappears in the linearized problem.

The equilibrium family is stable via center manifold theory.

Abstract

We study the asymptotics for large time of solutions to a one dimensional parabolic evolution equation with non-standard measure-valued right hand side, that involves derivatives of the solution computed at a free boundary point. The problem is a particular case of a mean-field free boundary model proposed by Lasry-Lions on price formation and dynamic equilibria. The main step in the proof is based on the fact that the free boundary disappears in the linearized problem, thus can be treated as a perturbation through semigroup theory. This requires a delicate choice for the function spaces since higher regularity is needed near the free boundary. We show global existence for solutions with initial data in a small neighborhood of any equilibrium point, and exponential decay towards a stationary state. Moreover, the family of equilibria of the equation is stable, as follows from center manifold theory.