A regular equilibrium solves the extended HJB system

TL;DR

This paper establishes that solving the extended Hamilton-Jacobi-Bellman (HJB) system is both necessary and sufficient for finding equilibrium in time-inconsistent stochastic control problems modeled as intrapersonal games.

Contribution

It proves the necessity of solving the extended HJB system for equilibrium, extending previous sufficiency results under regularity assumptions.

Findings

Solving the extended HJB system is necessary for equilibrium.

The controlled process is modeled as a general Itô diffusion.

The results apply to time-inconsistent stochastic control problems.

Abstract

Control problems not admitting the dynamic programming principle are known as time-inconsistent. The game-theoretic approach is to interpret such problems as intrapersonal dynamic games and look for subgame perfect Nash equilibria. A fundamental result of time-inconsistent stochastic control is a verification theorem saying that solving the extended HJB system is a sufficient condition for equilibrium. We show that solving the extended HJB system is a necessary condition for equilibrium, under regularity assumptions. The controlled process is a general It\^o diffusion.