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

Yunfei Peng, Wei Wei

TL;DR

This work addresses time-inconsistent deterministic linear-quadratic control under non-exponential discounting by formulating equilibrium HJB equations within an intra-personal game framework. It develops a novel method that connects the equilibrium value function to a Riccati equation with integral terms through the second derivative $D_x^2 V$, enabling a Riccati-based characterization of equilibria. A central result is the uniqueness of the equilibrium HJB solution, proven via the uniqueness of the associated Riccati equation and auxiliary ODEs, thereby establishing a well-posed equilibrium framework for time-inconsistent LQ problems. The findings provide a solid theoretical foundation for equilibria in time-inconsistent control and extend classical LQ theory to settings with non-exponential discounting, with implications for broader applications in dynamic decision making.

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.

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

TL;DR

This work addresses time-inconsistent deterministic linear-quadratic control under non-exponential discounting by formulating equilibrium HJB equations within an intra-personal game framework. It develops a novel method that connects the equilibrium value function to a Riccati equation with integral terms through the second derivative , enabling a Riccati-based characterization of equilibria. A central result is the uniqueness of the equilibrium HJB solution, proven via the uniqueness of the associated Riccati equation and auxiliary ODEs, thereby establishing a well-posed equilibrium framework for time-inconsistent LQ problems. The findings provide a solid theoretical foundation for equilibria in time-inconsistent control and extend classical LQ theory to settings with non-exponential discounting, with implications for broader applications in dynamic decision making.

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.
Paper Structure (11 sections, 8 theorems, 105 equations)

This paper contains 11 sections, 8 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Problem setting
  3. The equilibrium Bellman principle and HJB equations
  4. The equilibrium Bellman principle
  5. The equilibrium HJB equation and verification
  6. Uniqueness of the solutions to the equilibrium HJB equation
  7. A Riccati equation
  8. Two auxiliary functions
  9. Characterization of the solution to the equilibrium HJB equation
  10. Uniqueness
  11. Concluding remarks

Key Result

Theorem 1

Suppose that Assumptions (H0)-(H5) hold and the equilibrium value function $V\in WC^{1,1}$$\left([0,T]\times \mathbb{R}^n;\mathbb{R}\right)$, then the equilibrium value function $V$ satisfies the following equilibrium Bellman principle $\forall (t,x)\in [0,T]\times \mathbb{R}^n$ and $s\in[t,T]$.

Theorems & Definitions (14)

  • Definition 1
  • Definition 2: bjomur2009
  • Remark 1
  • Theorem 1
  • Theorem 2
  • Remark 2
  • Definition 3
  • Definition 4
  • Proposition 1
  • Proposition 2
  • ...and 4 more