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Mean-field stochastic linear quadratic control problem with random coefficients

Jie Xiong, Wen Xu

TL;DR

This work addresses MFSLQ control with random coefficients by first proving existence and uniqueness of an optimal control and deriving a stochastic maximum principle. The main challenge is a non-separable adjoint term $E[A_1^T Y]$ in the SMP, which the authors overcome by decomposing the problem into two mean-field–free SLQ problems and applying an extended LaGrange multiplier (ELM) method to handle a constraint on $E[X]$. The constrained problem is solved via invariant embedding and stochastic Riccati equations, yielding explicit representations for the optimal control in terms of linear operators and a deterministic mean-field quantity $\alpha^*(\cdot)$, which itself is found from a linear operator equation. The resulting framework provides a constructive, feedback-like solution to MFSLQ with random coefficients and suggests broad applicability to other mean-field control problems and extensions to conditional mean-field LQ settings.

Abstract

In this paper, we first prove that the mean-field stochastic linear quadratic (MFSLQ for short) control problem with random coefficients has a unique optimal control and derive a preliminary stochastic maximum principle to characterize this optimal control by an optimality system. However, because of the term of the form $\mathbb{E}[A_1(\cdot)^\top Y(\cdot)] $ in the adjoint equation, which cannot be represented in the form $\mathbb{E}[A_1(\cdot)^\top]\mathbb{E} [Y(\cdot)] $, we cannot solve this optimality system explicitly. To this end, we decompose the MFSLQ control problem into two problems without the mean-field terms, and one of them is a constrained problem. The constrained SLQ control problem is solved explicitly by an extended LaGrange multiplier method developed in this article.

Mean-field stochastic linear quadratic control problem with random coefficients

TL;DR

This work addresses MFSLQ control with random coefficients by first proving existence and uniqueness of an optimal control and deriving a stochastic maximum principle. The main challenge is a non-separable adjoint term in the SMP, which the authors overcome by decomposing the problem into two mean-field–free SLQ problems and applying an extended LaGrange multiplier (ELM) method to handle a constraint on . The constrained problem is solved via invariant embedding and stochastic Riccati equations, yielding explicit representations for the optimal control in terms of linear operators and a deterministic mean-field quantity , which itself is found from a linear operator equation. The resulting framework provides a constructive, feedback-like solution to MFSLQ with random coefficients and suggests broad applicability to other mean-field control problems and extensions to conditional mean-field LQ settings.

Abstract

In this paper, we first prove that the mean-field stochastic linear quadratic (MFSLQ for short) control problem with random coefficients has a unique optimal control and derive a preliminary stochastic maximum principle to characterize this optimal control by an optimality system. However, because of the term of the form in the adjoint equation, which cannot be represented in the form , we cannot solve this optimality system explicitly. To this end, we decompose the MFSLQ control problem into two problems without the mean-field terms, and one of them is a constrained problem. The constrained SLQ control problem is solved explicitly by an extended LaGrange multiplier method developed in this article.
Paper Structure (6 sections, 13 theorems, 98 equations)

This paper contains 6 sections, 13 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. A preliminary stochastic maximum principle
  4. Extended LaGrange multiplier method
  5. Optimal mean-field
  6. Conclusion

Key Result

Theorem 2.1

Let (H1) and (H2) hold. Then, Problem (MFSLQ) has a unique optimal control. Further, $u^\ast(\cdot)$ is the optimal control for Problem (MFSLQ) if and only if the adapted solution $(X^\ast(\cdot), Y^\ast(\cdot), Z^\ast(\cdot))\in$L^2,c_F(R^n)$^2\times L^2_\mathbb{F}(\mathbb{R}^n)$ to the following F admits the stationary condition:

Theorems & Definitions (22)

  • Theorem 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Remark 2.5
  • Remark 2.6
  • Lemma 2.7
  • Lemma 2.8
  • Lemma 2.9
  • Theorem 2.10
  • ...and 12 more