Communications in Mathematical Sciences

Volume 22 (2024)

Number 1

A variational approach for price formation models in one dimension

Pages: 227 – 255

DOI: https://dx.doi.org/10.4310/CMS.2024.v22.n1.a10

Authors

Yuri Ashrafyan (Computer, Electrical, Mathematical Sciences & Engineering (CEMSE), King Abdullah University of Science and Technology, Thuwal, Saudi Arabia)

Tigran Bakaryan (Computer, Electrical, Mathematical Sciences & Engineering (CEMSE), King Abdullah University of Science and Technology, Thuwal, Saudi Arabia)

Diogo Gomes (Computer, Electrical, Mathematical Sciences & Engineering (CEMSE), King Abdullah University of Science and Technology, Thuwal, Saudi Arabia)

Julian Gutierrez (Computer, Electrical, Mathematical Sciences & Engineering (CEMSE), King Abdullah University of Science and Technology, Thuwal, Saudi Arabia)

Abstract

In this paper, we study a class of first-order mean-field games (MFGs) that model price formation. Using Poincaré lemma, we eliminate one of the equations of the MFGs system and obtain a variational problem for a single function. We prove the uniqueness of the solutions to the variational problem and address the existence of solutions by applying relaxation arguments. Moreover, we establish a correspondence between solutions of the MFGs system and the variational problem. Based on this correspondence, we introduce an alternative numerical approach for the solution of the original MFGs problem. We end the paper with numerical results for a linear-quadratic model.

Keywords

mean field games, price formation, potential function, Lagrange multiplier

2010 Mathematics Subject Classification

35A15, 49N10

Received 30 May 2022

Received revised 8 June 2023

Accepted 8 June 2023

Published 7 December 2023