Time-Inconsistent Stochastic Linear--Quadratic Control

TL;DR

This paper introduces a framework for time-inconsistent stochastic linear-quadratic control problems, deriving equilibrium solutions and explicit strategies, especially in financial models with stochastic parameters.

Contribution

It formulates a general time-inconsistent stochastic LQ control problem, defines equilibrium controls, and derives explicit solutions in specific cases including financial applications.

Findings

Explicit equilibrium control in one-dimensional deterministic coefficient case

Sufficient conditions for equilibrium controls via forward-backward SDEs

Explicit strategies for mean-variance portfolio with stochastic risk premium

Abstract

In this paper, we formulate a general time-inconsistent stochastic linear--quadratic (LQ) control problem. The time-inconsistency arises from the presence of a quadratic term of the expected state as well as a state-dependent term in the objective functional. We define an equilibrium, instead of optimal, solution within the class of open-loop controls, and derive a sufficient condition for equilibrium controls via a flow of forward--backward stochastic differential equations. When the state is one dimensional and the coefficients in the problem are all deterministic, we find an explicit equilibrium control. As an application, we then consider a mean-variance portfolio selection model in a complete financial market where the risk-free rate is a deterministic function of time but all the other market parameters are possibly stochastic processes. Applying the general sufficient condition, we obtain explicit equilibrium strategies when the risk premium is both deterministic and stochastic.