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The mean-field control problem for heterogeneous forward-backward systems

Andreas Sojmark, Zeng Zhang

Abstract

We study the problem of mean-field control when the state dynamics are given by general systems of forward-backward stochastic differential equations (FBSDEs) with heterogeneous mean-field interactions. Firstly, we introduce a novel methodology for reducing the well-posedness of such systems to that of a single randomized mean-field FBSDE. As a consequence, we show that, in the fully coupled case, smallness conditions yield existence and uniqueness for both the system itself and the associated variational and adjoint systems. Secondly, we derive a stochastic maximum principle and a verification for the mean-field control problem. This provides necessary and sufficient conditions for optimality.

The mean-field control problem for heterogeneous forward-backward systems

Abstract

We study the problem of mean-field control when the state dynamics are given by general systems of forward-backward stochastic differential equations (FBSDEs) with heterogeneous mean-field interactions. Firstly, we introduce a novel methodology for reducing the well-posedness of such systems to that of a single randomized mean-field FBSDE. As a consequence, we show that, in the fully coupled case, smallness conditions yield existence and uniqueness for both the system itself and the associated variational and adjoint systems. Secondly, we derive a stochastic maximum principle and a verification for the mean-field control problem. This provides necessary and sufficient conditions for optimality.
Paper Structure (16 sections, 19 theorems, 127 equations)

This paper contains 16 sections, 19 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. The mean-field control problem
  3. Related literature and main contributions
  4. Structure of the paper
  5. Well-posedness of the mean-field FBSDE system
  6. Preliminaries and notation
  7. The FBSDE system
  8. Well-posedness
  9. The heterogeneous mean-field control problem
  10. Differentiability along the Markov kernels
  11. The maximum principle
  12. Verification Theorem
  13. Well-posedness of the FBSDE system
  14. Auxiliary FBSDE problem
  15. Existence and uniqueness
  16. ...and 1 more sections

Key Result

Lemma 2.2

Let $\phi:U\times[0,T]\times \mathbb{R}^n \times \mathbb{R}^l \times \mathbb{R}^{l\times d}\times \mathbb{R}^k \rightarrow [0,\infty)$ be Borel measurable. Then is $\mathrm{Leb}\otimes\mathcal{U}$-measurable.

Theorems & Definitions (40)

  • Definition 2.1: Strong solution
  • Remark 2.1
  • Lemma 2.2: Joint measurability
  • proof
  • Theorem 2.3
  • Remark 2.2
  • Theorem 2.4
  • Definition 3.1: $L_m$-differentiability
  • Remark 3.1
  • Theorem 3.2
  • ...and 30 more