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Linear-quadratic control for mean-field backward stochastic differential equations with random coefficients

Jie Xiong, Wen Xu, Ying Yang

Abstract

In this paper, we study the linear-quadratic control problem for mean-field backward stochastic differential equations (MF-BSDE) with random coefficients. We first derive a preliminary stochastic maximum principle to analyze the unique solvability of the optimality system for this control problem through the variational method. Subsequently, we reformulate the mean-field linear-quadratic (MF-BSLQ) problem as a constrained BSDE control problem by imposing constraints on the expectation processes, which we solve using the Extended Lagrange multiplier method. Finally, we derive an explicit expression for the optimal control associated with Problem (MF-BSLQ).

Linear-quadratic control for mean-field backward stochastic differential equations with random coefficients

Abstract

In this paper, we study the linear-quadratic control problem for mean-field backward stochastic differential equations (MF-BSDE) with random coefficients. We first derive a preliminary stochastic maximum principle to analyze the unique solvability of the optimality system for this control problem through the variational method. Subsequently, we reformulate the mean-field linear-quadratic (MF-BSLQ) problem as a constrained BSDE control problem by imposing constraints on the expectation processes, which we solve using the Extended Lagrange multiplier method. Finally, we derive an explicit expression for the optimal control associated with Problem (MF-BSLQ).

Paper Structure

This paper contains 6 sections, 13 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results
  4. The proof to Theorem \ref{['preSMP']}
  5. Solving Problem (a)
  6. The proof to Theorem \ref{['main']}

Key Result

Theorem 2.1

Let (H1) hold. Then, for any terminal state $\xi \in L^2_{\mathcal{F}_T}(\mathbb{R}^n)$ and control $u(\cdot)\in \mathcal{U}[0,T]$, the state equation $(state1)$ admits a unique solution Moreover, there exists a constant $K >0$, which is independent of $\xi$ and $u(\cdot)$ such that

Theorems & Definitions (18)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 3.1
  • Remark 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Remark 3.5
  • Lemma 3.6
  • Lemma 3.7
  • Lemma 3.8
  • ...and 8 more