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 $\varepsilon_N$-Nash equilibrium for the $N$-player game, with an explicit rate as the number of agents tends to infinity.