{"title":"A \\(C^1\\)-Itô’s Formula for Flows of Semimartingale Distributions","authors":"Bruno Bouchard, Xiaolu Tan, Jixin Wang","doi":"10.1007/s00245-024-10165-y","DOIUrl":null,"url":null,"abstract":"<div><p>We provide an Itô’s formula for <span>\\(C^1\\)</span>-functionals of flows of conditional marginal distributions of continuous semimartingales. This is based on the notion of weak Dirichlet process, and extends the <span>\\(C^1\\)</span>-Itô’s formula in Gozzi and Russo (Stoch Process Appl 116(11):1563–1583, 2006) to this context. As the first application, we study a class of McKean–Vlasov optimal control problems, and establish a verification theorem which only requires <span>\\(C^1\\)</span>-regularity of its value function, which is equivalently the (viscosity) solution of the associated HJB master equation. It goes together with a novel duality result.</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"90 1","pages":""},"PeriodicalIF":1.6000,"publicationDate":"2024-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00245-024-10165-y.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Applied Mathematics and Optimization","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00245-024-10165-y","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
We provide an Itô’s formula for \(C^1\)-functionals of flows of conditional marginal distributions of continuous semimartingales. This is based on the notion of weak Dirichlet process, and extends the \(C^1\)-Itô’s formula in Gozzi and Russo (Stoch Process Appl 116(11):1563–1583, 2006) to this context. As the first application, we study a class of McKean–Vlasov optimal control problems, and establish a verification theorem which only requires \(C^1\)-regularity of its value function, which is equivalently the (viscosity) solution of the associated HJB master equation. It goes together with a novel duality result.
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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