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Solvability of Equilibrium Riccati Equations: A Direct Approach

Bowen Ma, Hanxiao Wang

TL;DR

The paper tackles the solvability of the equilibrium Riccati equation (ERE) associated with closed-loop equilibrium strategies in time-inconsistent stochastic linear-quadratic problems. It introduces a direct approach that avoids the dynamic programming principle by establishing a priori estimates for the ERE with smooth coefficients and solving via a Picard iteration, yielding both local and global solvability and a practical numerical scheme; a mollification technique extends results to non-smooth data. A key insight is the equivalence between the ERE and a Volterra integro-differential equation for the diagonal P(t,t), enabling contraction-based fixed-point arguments. This work strengthens the theoretical foundation for equilibrium controls in time-inconsistent settings and provides implementable methods for computing the equilibrium strategy. The results connect to broader literature on time-inconsistent control, equilibrium HJB concepts, and numerical schemes for Riccati-type equations.

Abstract

The solvability of equilibrium Riccati equations (EREs) plays a central role in the study of time-inconsistent stochastic linear-quadratic optimal control problems, because it paves the way to constructing a closed-loop equilibrium strategy. Under the standard conditions, Yong [29] established its well-posedness by introducing the well-known multi-person differential game method. However, this method depends on the dynamic programming principle (DPP) of the sophisticated problems on every subinterval, and thus is essentially a control theory approach. In this paper, we shall give a new and more direct proof, in which the DPP is no longer needed. We first establish a priori estimates for the ERE in the case of smooth coefficients. Using this estimate, we then demonstrate both the local and global solvability of the ERE by constructing an appropriate Picard iteration sequence, which actually provides a numerical algorithm. Additionally, a mollification method is employed to handle the case with non-smooth coefficients.

Solvability of Equilibrium Riccati Equations: A Direct Approach

TL;DR

The paper tackles the solvability of the equilibrium Riccati equation (ERE) associated with closed-loop equilibrium strategies in time-inconsistent stochastic linear-quadratic problems. It introduces a direct approach that avoids the dynamic programming principle by establishing a priori estimates for the ERE with smooth coefficients and solving via a Picard iteration, yielding both local and global solvability and a practical numerical scheme; a mollification technique extends results to non-smooth data. A key insight is the equivalence between the ERE and a Volterra integro-differential equation for the diagonal P(t,t), enabling contraction-based fixed-point arguments. This work strengthens the theoretical foundation for equilibrium controls in time-inconsistent settings and provides implementable methods for computing the equilibrium strategy. The results connect to broader literature on time-inconsistent control, equilibrium HJB concepts, and numerical schemes for Riccati-type equations.

Abstract

The solvability of equilibrium Riccati equations (EREs) plays a central role in the study of time-inconsistent stochastic linear-quadratic optimal control problems, because it paves the way to constructing a closed-loop equilibrium strategy. Under the standard conditions, Yong [29] established its well-posedness by introducing the well-known multi-person differential game method. However, this method depends on the dynamic programming principle (DPP) of the sophisticated problems on every subinterval, and thus is essentially a control theory approach. In this paper, we shall give a new and more direct proof, in which the DPP is no longer needed. We first establish a priori estimates for the ERE in the case of smooth coefficients. Using this estimate, we then demonstrate both the local and global solvability of the ERE by constructing an appropriate Picard iteration sequence, which actually provides a numerical algorithm. Additionally, a mollification method is employed to handle the case with non-smooth coefficients.
Paper Structure (4 sections, 6 theorems, 32 equations)

This paper contains 4 sections, 6 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Literature review on some related topics
  3. Preliminary
  4. A priori estimates

Key Result

Lemma 2.1

Suppose that ${\cal R}(\cdot)\in C([0,T];\,{\mathbb{S}}^{n })$ satisfies the following monotonic condition: Then ${\cal R}(\cdot)$ is differentiable almost everywhere and its derivative is positive semi-definite.

Theorems & Definitions (13)

  • Definition 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Remark 2.1
  • Lemma 2.4
  • proof
  • Remark 2.2
  • ...and 3 more