Robust Time-inconsistent Linear-Quadratic Stochastic Controls: A Stochastic Differential Game Approach

TL;DR

This paper develops a stochastic differential game approach to robust time-inconsistent linear-quadratic control problems, deriving conditions for Nash equilibrium and illustrating the impact of ambiguity aversion on risk behavior.

Contribution

It introduces a novel framework for analyzing robust TIC control problems using stochastic differential games and provides numerical insights into ambiguity effects.

Findings

Ambiguity aversion accelerates risk aversion in TIC controls.

The framework derives sufficient conditions for Nash equilibrium in TIC problems.

Ambiguity impacts are more significant in TIC cases than in TC counterparts.

Abstract

This paper studies robust time-inconsistent (TIC) linear-quadratic stochastic control problems, formulated by stochastic differential games. By a spike variation approach, we derive sufficient conditions for achieving the Nash equilibrium, which corresponds to a time-consistent (TC) robust policy, under mild technical assumptions. To illustrate our framework, we consider two scenarios of robust mean-variance analysis, namely with state- and control-dependent ambiguity aversion. We find numerically that with time inconsistency haunting the dynamic optimal controls, the ambiguity aversion enhances the effective risk aversion faster than the linear, implying that the ambiguity in the TIC cases is more impactful than that under the TC counterparts, e.g., expected utility maximization problems.