Unified continuous-time q-learning for mean-field game and mean-field control problems

TL;DR

This paper introduces a unified continuous-time q-learning framework for mean-field game and control problems in jump-diffusion models, enabling policy evaluation and learning without direct population distribution observation.

Contribution

It proposes the decoupled Iq-function with martingale characterization, unifying policy evaluation for MFG and MFC, and develops a novel q-learning algorithm applicable to financial jump-diffusion models.

Findings

Successful application to financial jump-diffusion models

Exact parameterization of Iq-functions and value functions

Satisfactory performance of the proposed q-learning algorithm

Abstract

This paper studies the continuous-time q-learning in mean-field jump-diffusion models when the population distribution is not directly observable. We propose the integrated q-function in decoupled form (decoupled Iq-function) from the representative agent's perspective and establish its martingale characterization, which provides a unified policy evaluation rule for both mean-field game (MFG) and mean-field control (MFC) problems. Moreover, we consider the learning procedure where the representative agent updates the population distribution based on his own state values. Depending on the task to solve the MFG or MFC problem, we can employ the decoupled Iq-function differently to characterize the mean-field equilibrium policy or the mean-field optimal policy respectively. Based on these theoretical findings, we devise a unified q-learning algorithm for both MFG and MFC problems by utilizing test policies and the averaged martingale orthogonality condition. For several financial applications in the jump-diffusion setting, we obtain the exact parameterization of the decoupled Iq-functions and the value functions, and illustrate our q-learning algorithm with satisfactory performance.