Time-inconsistent stochastic linear-quadratic control in continuous time

Abstract

This paper studies a process of dealing with time inconsistent stochastic control problems using a system of Hamilton-Jacobi-Bellman equations. Such an approach aims to obtain an equilibrium strategy—through a subgame perfect Nash Equilibrium perspective—from which does not necessarily maximise the objective at every point. However, the time consistency of the strategy allows for a more practical resort considering a time inconsistent environment. Specifically, this study examines the application of the framework on a class of general linear-quadratic control problems. This class of linear-quadratic control problems contains an interaction term between present state and expectations— which is one of the ways of generating state dependency. The aim of this study is to obtain more explicit semi-closed forms of equations that are dependent on the stated parameters. The study developed a direct and straightforward procedure to derive the solutions for coefficient terms of the equilibrium value function and equilibrium control law. In the process, it also outlines a manoeuvre to manage and simplify layers of derivatives and integrals. The findings are then applied to evaluate a portfolio maximisation problem. The portfolio consists of only risky assets and assumes a self-financing constraint. The objective function considers a mean-variance problem with state dependent risk aversion.