Instability and bifurcation in a trend depending price formation model

TL;DR

This paper modifies a classical market price formation model to demonstrate how certain parameter values can lead to instabilities, resulting in oscillatory and wave-like price behaviors such as inflation or deflation.

Contribution

The paper introduces modifications to the Lasry-Lions model to reveal instabilities and bifurcations, including periodic solutions and traveling waves, capturing complex market dynamics.

Findings

Identification of parameter ranges causing instability

Existence of periodic solutions via Hopf bifurcation

Presence of traveling wave solutions indicating inflation or deflation

Abstract

A well-known model due to J.-M. Lasry and P.L. Lions that presents the evolution of prices in a market as the evolution of a free boundary in a diffusion equation is suggested to be modified in order to show instabilities for some values of the parameters. This loss of stability is associated to the appearance of new types of solutions, namely periodic solutions, due to a Hopf bifurcation and representing price oscillations, and traveling waves, that represent either inflationary or deflationary behavior.