Time Inconsistent Stochastic Control in Continuous Time: Theory and Examples

TL;DR

This paper develops a continuous-time stochastic control theory addressing time inconsistency, using game theory to find Nash equilibrium strategies and extending Hamilton-Jacobi-Bellman equations for various applications.

Contribution

It introduces a novel framework for continuous-time time-inconsistent control problems using game theory and derives a system of equations for equilibrium strategies.

Findings

Extended HJB equations for equilibrium strategies

Application to non exponential discounting and mean variance problems

Analysis of time inconsistency in economic models

Abstract

In this paper, which is a continuation of the previously published discrete time paper we develop a theory for continuous time stochastic control problems which, in various ways, are time inconsistent in the sense that they do not admit a Bellman optimality principle. We study these problems within a game theoretic framework, and we look for Nash subgame perfect equilibrium points. For a general controlled continuous time Markov process and a fairly general objective functional we derive an extension of the standard Hamilton-Jacobi-Bellman equation, in the form of a system of non-linear equations, for the determination for the equilibrium strategy as well as the equilibrium value function. As applications of the general theory we study non exponential discounting, various types of mean variance problems, a point process example, as well as a time inconsistent linear quadratic regulator. We also present a study of time inconsistency within the framework of a general equilibrium production economy of Cox-Ingersoll-Ross type.