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

Paper: arXiv 2306.16982

Authors: Bingyan Han, Chi Seng Pun, Hoi Ying Wong

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.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 2.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a highly sophisticated theoretical framework using stochastic differential games, spike variations, and linear-quadratic control to address time-inconsistent robust control. However, it relies primarily on numerical illustrations rather than empirical data validation or real-world financial backtesting.

Research Flowchart

  flowchart TD
    A[Research Goal: Robust Time-Inconsistent LQ Stochastic Control] --> B[Formulate Framework as Stochastic Differential Game]
    B --> C[Inputs: State- & Control-Dependent Ambiguity Aversion]
    C --> D[Methodology: Apply Spike Variation Approach]
    D --> E[Derive Conditions for Nash Equilibrium / TC Policy]
    E --> F[Numerical Analysis of Robust Mean-Variance Problems]
    F --> G[Finding: Ambiguity aversion enhances risk aversion faster in TIC vs TC cases]