A Progressive Maximum Principle of Fully Coupled Mean-Field System with Jumps

Tian Chen (), Hongyu Shi () and Zhen Wu ()
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Tian Chen: Suzhou Research Institute, Shandong University
Hongyu Shi: School of Mathematics, Shandong University
Zhen Wu: School of Mathematics, Shandong University

Journal of Optimization Theory and Applications, 2025, vol. 206, issue 3, No 19, 28 pages

Abstract: Abstract In this paper, we investigate a class of progressive optimal control problems for fully coupled mean-field forward-backward systems with random jumps. Under weakly-coupled conditions and an arbitrary fixed time horizon, we establish the well-posedness of a class of fully coupled mean-field forward-backward stochastic differential equations with jumps, ensuring the well-posedness of the state, variational and adjoint equations. Next, using the convex variational method, we provide a stochastic maximum principle for the progressive optimal control of this mean-field system. Our maximum principle is divided into two parts: a continuous component, which characterizes the optimal control during continuous periods, and a jump component, which defines the optimal control behavior at jump times. Additionally, we provide a sufficient maximum principle under certain convexity assumptions. Finally, we apply these theoretical results to a linear-quadratic control problem, obtaining both the open-loop optimal control and its corresponding feedback representation, further demonstrating the practical effectiveness of our findings.

Keywords: Fully coupled forward-backward stochastic differential equation; Progressive structure; Mean-field; Stochastic maximum principle; 93E20; 49N80; 65C99 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s10957-025-02760-y

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