{"title":"Stochastic Linear-Quadratic Optimal Control Problems with Multi-dimensional State, Random Coefficients and Regime Switching","authors":"Yuyang Chen, Peng Luo","doi":"10.1007/s00245-025-10235-9","DOIUrl":null,"url":null,"abstract":"<div><p>This paper investigates the stochastic linear-quadratic (LQ, for short) optimal control problems with random coefficients and regime switching in a finite time horizon where the state equation is multi-dimensional. Similar to the classical stochastic LQ problems, we establish the relationship between the stochastic LQ optimal control problems with regime switching and the related extended stochastic Riccati equations. To solve the extended stochastic Riccati equations, we construct a monotone Piccard iterative sequence and present the link between this sequence and solutions of a family of forward-backward stochastic differential equations. Relying on <span>\\(L^p\\)</span> estimates for FBSDEs, we show that the extended stochastic Riccati equation has a solution. This partially addresses one question left in Hu et al. (Ann. Appl. Probab. 32(1): 426-460, 2022). Finally, the stochastic LQ optimal control problems with regime switching is solved.</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"91 2","pages":""},"PeriodicalIF":1.6000,"publicationDate":"2025-03-05","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/s00245-025-10235-9","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 the stochastic linear-quadratic (LQ, for short) optimal control problems with random coefficients and regime switching in a finite time horizon where the state equation is multi-dimensional. Similar to the classical stochastic LQ problems, we establish the relationship between the stochastic LQ optimal control problems with regime switching and the related extended stochastic Riccati equations. To solve the extended stochastic Riccati equations, we construct a monotone Piccard iterative sequence and present the link between this sequence and solutions of a family of forward-backward stochastic differential equations. Relying on \(L^p\) estimates for FBSDEs, we show that the extended stochastic Riccati equation has a solution. This partially addresses one question left in Hu et al. (Ann. Appl. Probab. 32(1): 426-460, 2022). Finally, the stochastic LQ optimal control problems with regime switching is solved.
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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