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Abstract <p>When describing the group behavior of high-frequency traders, there arises a boundary value problem 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 distribution density of traders in forward time. The system is inherently ill-conditioned due to the turnpike effect. Under certain assumptions, it is possible to perform reduction to a system of Riccati equations; however, the question of well-posedness of the reduced problem remains open. This work investigates this question, namely, the conditions for the existence and uniqueness of the solution to the boundary value problem depending on the model parameters.</p>

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Studying the Well-Posedness of the Boundary Value Problem for a System of Riccati Type Equations Based on the Concept of Mean Field Games

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  • 发表日期：

    2025年6月16日

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  • F. A. Fedorov

摘要

Abstract

When describing the group behavior of high-frequency traders, there arises a boundary value problem 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 distribution density of traders in forward time. The system is inherently ill-conditioned due to the turnpike effect. Under certain assumptions, it is possible to perform reduction to a system of Riccati equations; however, the question of well-posedness of the reduced problem remains open. This work investigates this question, namely, the conditions for the existence and uniqueness of the solution to the boundary value problem depending on the model parameters.