{"title":"Linear-Quadratic Mean-Field Backward Stackelberg Game with Mixed Terminal Perturbation","authors":"Tian Chen, Xinwei Feng, Yunxiao Jia","doi":"10.1007/s00245-025-10225-x","DOIUrl":null,"url":null,"abstract":"<div><p>This paper investigates a linear-quadratic (LQ) Stackelberg game for mean-field type backward stochastic system, in which the cost functional is also mean-field type. In our model, the leader first announces the terminal goal satisfying pointwise and affine constraints and open-loop dynamic decisions at the initial time which takes into account the best response of the follower. Then two interrelated optimization problems are sequentially solved by the follower (a backward LQ problem) and the leader (a backward-forward LQ problem). The open-loop Stackelberg equilibrium is obtained by virtue of duality theory and represented by some fully coupled mean-field backward-forward stochastic differential equations with mixed initial-terminal conditions, whose global solvability is discussed in some nontrivial cases by Riccati decoupling method and discounting method. As an application, we address a product pricing problem.</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"91 2","pages":""},"PeriodicalIF":1.7000,"publicationDate":"2025-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Applied Mathematics and Optimization","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00245-025-10225-x","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
This paper investigates a linear-quadratic (LQ) Stackelberg game for mean-field type backward stochastic system, in which the cost functional is also mean-field type. In our model, the leader first announces the terminal goal satisfying pointwise and affine constraints and open-loop dynamic decisions at the initial time which takes into account the best response of the follower. Then two interrelated optimization problems are sequentially solved by the follower (a backward LQ problem) and the leader (a backward-forward LQ problem). The open-loop Stackelberg equilibrium is obtained by virtue of duality theory and represented by some fully coupled mean-field backward-forward stochastic differential equations with mixed initial-terminal conditions, whose global solvability is discussed in some nontrivial cases by Riccati decoupling method and discounting method. As an application, we address a product pricing problem.
期刊介绍：
The Applied Mathematics and Optimization Journal covers a broad range of mathematical methods in particular those that bridge with optimization and have some connection with applications. Core topics include calculus of variations, partial differential equations, stochastic control, optimization of deterministic or stochastic systems in discrete or continuous time, homogenization, control theory, mean field games, dynamic games and optimal transport. Algorithmic, data analytic, machine learning and numerical methods which support the modeling and analysis of optimization problems are encouraged. Of great interest are papers which show some novel idea in either the theory or model which include some connection with potential applications in science and engineering.
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