Here, there and everywhere: state-dependent time-inconsistent stochastic control

TL;DR

This paper develops a probabilistic framework using backward stochastic differential equations to address time-inconsistent stochastic control problems in continuous time, exemplified by a linear-quadratic regulator application.

Contribution

It introduces a novel BSDE-based representation for equilibrium value functions in time-inconsistent control, advancing theoretical understanding and solution methods.

Findings

Derived a system of BSDEs characterizing equilibrium controls

Applied the framework to a linear-quadratic regulator example

Provided insights into handling state-dependent time inconsistency

Abstract

This paper addresses the challenge of time-inconsistent stochastic control within a continuous-time framework. Its primary focus lies in uncovering a probabilistic representation, specifically in the shape of a system of backward stochastic differential equations (BSDEs). These equations encapsulate the equilibrium value function essential for resolving cases where the present state affecting the target functional triggers the inconsistency. Additionally, the paper offers an application exemplifying this theory through the time-inconsistent linear--quadratic regulator.