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

TL;DR

This paper proves the uniqueness and continuity of the optimal boundary surface in a mean-field game of capacity expansion, characterizes it via a nonlinear integral equation, and provides a numerical algorithm for solutions.

Contribution

It establishes the unique continuous solution of the boundary surface as a nonlinear integral equation and extends Itô's formula for better numerical analysis.

Findings

Proved the optimal boundary surface is the unique continuous solution of a Volterra-type integral equation.

Developed an extension of Itô's formula with weaker assumptions.

Provided a numerical algorithm and computed optimal controls for the mean-field game.

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.