Solutions to Equilibrium HJB Equations for Time-Inconsistent Deterministic Linear Quadratic Control: Characterization and Uniqueness

Paper: arXiv 2308.13850

Authors: Yunfei Peng, Wei Wei

Abstract

In this paper we study a class of HJB equations which solve for equilibria for general time-inconsistent deterministic linear quadratic control problems within the intra-personal game theoretic framework, where the inconsistency arises from non-exponential discount functions. We characterize the solutions to the HJB equations using a class of Riccati equations with integral terms. By studying the uniqueness of solutions to the integro-differential Riccati equations, we prove the uniqueness of solutions to the equilibrium HJB equations.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 1.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper presents a rigorous theoretical analysis of equilibrium HJB equations and non-standard integro-differential Riccati equations under non-exponential discounting. However, it is purely theoretical with no empirical backtesting, datasets, or real-world market applications included.

Research Flowchart

  flowchart TD
    A["Research Goal: Solve & Prove Uniqueness for<br>Equilibrium HJB Equations in Time-Inconsistent LQ Control"] --> B["Inputs: Deterministic LQ System &<br>Non-Exponential Discount Functions"]
    B --> C["Framework: Intra-Personal Game Theory"]
    C --> D["Methodology: Characterize HJB Equations via<br>Integro-Differential Riccati Equations"]
    D --> E["Computational Analysis: Analyze Existence &<br>Uniqueness of Riccati Solutions"]
    E --> F["Key Outcome: Proof of Unique Solutions<br>for Equilibrium HJB Equations"]