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General mean-field stochastic linear quadratic control problem driven by Lévy processes with random coefficients

Yanyan Tang, Jie Xiong

TL;DR

This work tackles a stochastic mean-field linear-quadratic control problem with random coefficients and jumps driven by Brownian motion and a Poisson random measure. The authors develop an extended Lagrange multiplier approach to decouple the mean-field constraints, solving a constrained SLQ problem (Sub1) to obtain a state-feedback control $u^* = \Theta^{-1} N X^* + \Theta^{-1} M$ via a stochastic Riccati equation with jumps and a linear BSDE, and a dual problem (Sub2) to determine the optimal mean-field targets $(a^*,b^*)$ through operator equations. A key technical achievement is the decoupling of the fully coupled FBSDE system using the ansatz $Y = P X + \varphi$, where $P$ solves a stochastic Riccati equation with jumps and $(\varphi,\psi,\theta)$ solves a linear BSDE with jumps, ensuring a rigorous state-feedback representation. The combination of Sub1 and Sub2 yields a complete solution to the MF-SLQ problem in the Lévy-noise setting, extending prior homogeneous/deterministic coefficient results and providing explicit computable expressions for the optimal controller in terms of Riccati-type backward equations and linear operator mappings. This contributes a systematic framework for MF-SLQ control with random coefficients and jumps and offers a practical pathway to implement optimal feedback laws in systems subject to mean-field interactions and discontinuous disturbances.

Abstract

This paper studies a stochastic mean-field linear-quadratic optimal control problem with random coefficients. The state equation is a general linear stochastic differential equation with mean-field terms $\EE X(t)$ and $\EE u(t)$ of the state and the control processes and is driven by a Brownian motion and a Poisson random measure. By the coupled system of Riccati equations, an explicit expressions for the optimal state feedback control is obtained. As a by-product, the non-homogeneous stochastic linear-quadratic control problem with random coefficients and Lévy driving noises is also studied.

General mean-field stochastic linear quadratic control problem driven by Lévy processes with random coefficients

TL;DR

This work tackles a stochastic mean-field linear-quadratic control problem with random coefficients and jumps driven by Brownian motion and a Poisson random measure. The authors develop an extended Lagrange multiplier approach to decouple the mean-field constraints, solving a constrained SLQ problem (Sub1) to obtain a state-feedback control via a stochastic Riccati equation with jumps and a linear BSDE, and a dual problem (Sub2) to determine the optimal mean-field targets through operator equations. A key technical achievement is the decoupling of the fully coupled FBSDE system using the ansatz , where solves a stochastic Riccati equation with jumps and solves a linear BSDE with jumps, ensuring a rigorous state-feedback representation. The combination of Sub1 and Sub2 yields a complete solution to the MF-SLQ problem in the Lévy-noise setting, extending prior homogeneous/deterministic coefficient results and providing explicit computable expressions for the optimal controller in terms of Riccati-type backward equations and linear operator mappings. This contributes a systematic framework for MF-SLQ control with random coefficients and jumps and offers a practical pathway to implement optimal feedback laws in systems subject to mean-field interactions and discontinuous disturbances.

Abstract

This paper studies a stochastic mean-field linear-quadratic optimal control problem with random coefficients. The state equation is a general linear stochastic differential equation with mean-field terms and of the state and the control processes and is driven by a Brownian motion and a Poisson random measure. By the coupled system of Riccati equations, an explicit expressions for the optimal state feedback control is obtained. As a by-product, the non-homogeneous stochastic linear-quadratic control problem with random coefficients and Lévy driving noises is also studied.

Paper Structure

This paper contains 8 sections, 16 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results
  4. A preliminary stochastic maximum principle
  5. Extened LaGrange multiplier method
  6. Optimize the control variable for fixed ELMs
  7. Optimize the ELMs
  8. Optimal mean-field

Key Result

Theorem 2.1

Consider the MF-SDE with jumps where $a$ is an ${\cal F}_0$-measurable random variable and $b, {\sigma}:[0,T]\times\Omega\times{\fam\msbmfam R}^n\times{\fam\msbmfam R}^n\times {\fam\msbmfam R}^m\times{\fam\msbmfam R}^m\to {\fam\msbmfam R}^n$, $\eta:[0,T]\times\Omega\times{\fam\msbmfam R}^n\times{\fam\msbmfam R}^n\times {\fam\msbm Then, for any $u\in L_{{\fam\msbmfam F}}^2(0,T; {\fam\msbmfam R}^m)

Theorems & Definitions (18)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 3.1
  • Theorem 3.2
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Lemma 3.6
  • ...and 8 more