A singular infinite dimensional Hamilton-Jacobi-Bellman equation arising from a storage problem

TL;DR

This paper derives and studies a complex infinite-dimensional Hamilton-Jacobi-Bellman equation from a storage model with many agents, addressing new mathematical challenges in its analysis.

Contribution

It introduces a novel infinite-dimensional HJB equation from a storage model with mean field game structure and analyzes its mathematical properties.

Findings

Derived an infinite-dimensional PDE for economic equilibrium.

Identified mathematical difficulties due to singularity and Hilbert space setting.

Provided insights into the structure of the mean field game master equation.

Abstract

In the first part of this paper, we derive an infinite dimensional partial differential equation which describes an economic equilibrium in a model of storage which includes an infinite number of non-atomic agents. This equation has the form of a mean field game master equation. The second part of the paper is devoted to the mathematical study of the Hamilton-Jacobi-Bellman equation from which the previous equation derives. This last equation is both singular and set on a Hilbert space and thus raises new mathematical difficulties.