Investigation of the well-posedness of the boundary value problem for a system of Riccati-type equations based on the mean-field games framework

Recieved: 11/15/2024

Accepted: 11/26/2024

Published: 06/20/2025

Keywords: mean field games, a system of Riccati equations, a boundary value problem for a system of ODEs

To cite this article

Fedorov F.A. Investigation of the well-posedness of the boundary value problem for a system of Riccati-type equations based on the mean-field games framework. // Moscow University Journal. Series 15. Computational Mathematics and Cybernetics. 2025. N 2, p.69-82 https://doi.org/10.55959/MSU/0137–0782–15–2025–49–2–69–82.

This work is licensed under a Сreative Commons Atribiution - NonCommercial 4.0 International (CC BY-NC 4.0)

Fedorov F.A. (Federal Research Center "Science and Management", Russian Academy of Sciences, Moscow, Russia)

Abstract

When describing the group behavior of high-frequency traders, a boundary value problem arises based on the concept of mean field games. The system consists of two coupled partial differential equations: the Hamilton–Jacobi–Bellman equation, which describes the evolution of the average payoff function in backward time, and the Kolmogorov–Fokker–Planck equation, which describes the evolution of the probability density distribution of traders in forward time. The system is ill-conditioned due to the turnpike effect. Under certain assumptions, it is possible to reduce the system to a set of Riccati equations; however, the question of the well-posedness of the reduced problem remains open. This work investigates this question, specifically the conditions for the existence and uniqueness of the solution to the boundary value problem depending on the model parameters.