Time-Inconsistent Optimal Control Problems and the Equilibrium HJB Equation

TL;DR

This paper develops a framework for solving time-inconsistent stochastic control problems by deriving an equilibrium HJB equation, analyzing its properties, and constructing time-consistent strategies, with applications to portfolio optimization.

Contribution

It introduces a Hamilton-Jacobi-Bellman type equation for time-inconsistent problems and studies its well-posedness and solutions, extending classical control theory.

Findings

Derived an equilibrium HJB equation for stochastic control

Established well-posedness and properties of the equation

Constructed time-consistent equilibrium strategies

Abstract

A general time-inconsistent optimal control problem is considered for stochastic differential equations with deterministic coefficients. Under suitable conditions, a Hamilton-Jacobi-Bellman type equation is derived for the equilibrium value function of the problem. Well-posedness and some properties of such an equation is studied, and time-consistent equilibrium strategies are constructed. As special cases, the linear-quadratic problem and a generalized Merton's portfolio problem are investigated.