Linear-quadratic mean field games with Volterra dynamics
Abstract
We study a class of linear-quadratic mean field games in which the state of each agent follows a stochastic Volterra equation of convolution type. The resulting dynamics are non-Markovian and account for memory effects through a completely monotone kernel, while the mean field interaction enters both the drift and diffusion coefficients. Exploiting a Markovian lifting of the Volterra dynamics, which restores the Markov property at the price of moving to an infinite-dimensional state space, we characterize the optimal response of a representative agent for a given population mean through a system of Riccati equations in Banach spaces, thereby obtaining a feedback representation of the optimal control. We then study the consistency condition defining the mean field equilibrium, which reduces to a forward-backward fixed point equation for the equilibrium mean: we prove existence and uniqueness of the equilibrium on short horizons and, when the control does not act on the diffusion coefficient, on arbitrary horizons under a monotonicity condition on the interaction. Finally, we consider the corresponding finite-player game and establish a quantitative propagation of chaos result. As a consequence, the feedback control of any mean field equilibrium, when implemented by each agent on its own state, yields an εN-Nash equilibrium for the N-player game, with an explicit rate as the number of agents tends to infinity.
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BibTeX
@misc{fontana2026linearquadratic,
title = {{Linear-quadratic mean field games with Volterra dynamics}},
author = {Niccolò Fontana and Mattia Martini and Huyên Pham},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.37622},
eprint = {2610.37622},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.37622},
}
RIS
TY - UNPB
TI - Linear-quadratic mean field games with Volterra dynamics
AU - Niccolò Fontana
AU - Mattia Martini
AU - Huyên Pham
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.37622
UR - https://arxiv.org/abs/2610.37622
AB - We study a class of linear-quadratic mean field games in which the state of each agent follows a stochastic Volterra equation of convolution type. The resulting dynamics are non-Markovian and account for memory effects through a completely monotone kernel, while the mean field interaction enters both the drift and diffusion coefficients. Exploiting a Markovian lifting of the Volterra dynamics, which restores the Markov property at the price of moving to an infinite-dimensional state space, we characterize the optimal response of a representative agent for a given population mean through a system of Riccati equations in Banach spaces, thereby obtaining a feedback representation of the optimal control. We then study the consistency condition defining the mean field equilibrium, which reduces to a forward-backward fixed point equation for the equilibrium mean: we prove existence and uniqueness of the equilibrium on short horizons and, when the control does not act on the diffusion coefficient, on arbitrary horizons under a monotonicity condition on the interaction. Finally, we consider the corresponding finite-player game and establish a quantitative propagation of chaos result. As a consequence, the feedback control of any mean field equilibrium, when implemented by each agent on its own state, yields an ε_N-Nash equilibrium for the N-player game, with an explicit rate as the number of agents tends to infinity.
ER -