ORIGINAL PAPERS

A Variational Formula of Forward-Backward Stochastic Differential System of Mean-Field Type with Observation Noise and Some Application

  • Meijiao Wang ,
  • Maoning Tang ,
  • Qiuhong Shi ,
  • Qingxin Meng
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  • 1. Business School, University of Shanghai for Science and Technology, Shanghai, 200093, China;
    2. Department of Mathematics, Huzhou University, Huzhou, 313000, Zhejiang, China

Received date: 2023-12-29

  Revised date: 2024-04-02

  Online published: 2026-02-11

Supported by

This research was supported by the Key Projects of Natural Science Foundation of Zhejiang Province, China (No. LZ22A010005), the National Natural Science Foundation of China (No. 12271158), and the Natural Science Foundation of Zhejiang Province, China (No. LY21A010001). The funding played no role in the design, execution, analysis, or interpretation of the study.

Abstract

This paper examines an optimal control problem for mean-field systems under partial observation. The state system is described by a controlled mean-field forward-backward stochastic differential equation that features correlated noises between the system and the observation. Furthermore, the observation coefficients are allowed to depend not only on the control process but also on its probability distribution. Assuming a convex control domain and allowing all coefficients of the systems to be random, we derive directly a variational formula for the cost function in a given control process direction in terms of the Hamiltonian and the associated adjoint system without relying on variational systems under standard assumptions on the coefficients. As an application, we present the necessary and sufficient conditions for the optimality of our control problem using Pontryagin’s maximum principle in a unified manner. The research provides insights into the optimization of mean-field systems under partial observation, which has practical implications for various applications.

Cite this article

Meijiao Wang , Maoning Tang , Qiuhong Shi , Qingxin Meng . A Variational Formula of Forward-Backward Stochastic Differential System of Mean-Field Type with Observation Noise and Some Application[J]. Communications on Applied Mathematics and Computation, 2026 , 8(1) : 269 -286 . DOI: 10.1007/s42967-024-00431-9

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