Integral equations for the optimal boundary surface of a mean-field game of capacity expansion

Abstract

We prove that the optimal boundary surface that splits the action and inaction regions in a mean-field game of capacity expansion studied in (Campi et al.,\ Ann.\ Appl.\ Probab.,\ {\bf 32}(5),\, pp.\,3674-3717, 2022) is the unique continuous solution of a nonlinear integral equation of Volterra type. In order to do that, we first establish continuity of the optimal surface. Then we develop an extension of It\^o's formula which weakens assumptions required in the existing literature on the first-order time-derivative and/or second-order space derivative of the value function. The paper also provides an algorithm for the numerical solution of the integral equation and we compute optimal controls numerically for the mean-field game.