Linear Quadratic Nash Systems and Master Equations in Hilbert Spaces
Daria Ghilli, Michele Ricciardi
TL;DR
This work develops a theory for linear-quadratic Nash systems and Master equations in possibly infinite-dimensional Hilbert spaces, with mean-field interaction entering the cost only through the mean. By exploiting this structure, the authors reduce the Nash system and the Master equation to coupled Riccati equations and backward linear evolutions on a Hilbert space, and prove existence and uniqueness of solutions for all time horizons. They demonstrate a global well-posedness theory via fixed-point arguments on small time intervals and a priori estimates, complemented by an explicit vintage capital application reformulated in a Hilbert space setting. The results extend infinite-dimensional mean-field game theory to a broad class of LQ problems with age- or path-dependent features, and lay groundwork for verification results and convergence to Master equations, with the vintage capital model as a concrete illustration of the framework.
Abstract
This paper aims to develop a theory for linear-quadratic Nash systems and Master equations in possibly infinite-dimensional Hilbert spaces. As a first step and motivated by the recent results in [31], we study a more general model in the linear quadratic case where the dependence on the distribution enters just in the objective functional through the mean. This property enables the Nash systems and the Master equation to be reduced to two systems of coupled Riccati equations and backward abstract evolution equations. We show that solutions for such systems exist and are unique for all time horizons, a result that is completely new in the literature in our setting. Finally, we apply the results to a vintage capital model, where capital depends on time and age, and the production function depends on the mean of the vintage capital.