通过不同方法对多尺度 McKean-Vlasov SDEs 进行扩散逼近

IF 2.4 2区 数学 Q1 MATHEMATICS
Wei Hong , Shihu Li , Xiaobin Sun
{"title":"通过不同方法对多尺度 McKean-Vlasov SDEs 进行扩散逼近","authors":"Wei Hong ,&nbsp;Shihu Li ,&nbsp;Xiaobin Sun","doi":"10.1016/j.jde.2024.09.012","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper, our objective is to investigate the diffusion approximation for multi-scale McKean-Vlasov stochastic differential equations. More precisely, we first establish the tightness of the law of <span><math><msub><mrow><mo>{</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>ε</mi></mrow></msup><mo>}</mo></mrow><mrow><mn>0</mn><mo>&lt;</mo><mi>ε</mi><mo>⩽</mo><mn>1</mn></mrow></msub></math></span> in <span><math><mi>C</mi><mo>(</mo><mo>[</mo><mn>0</mn><mo>,</mo><mi>T</mi><mo>]</mo><mo>;</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></math></span>. Subsequently, we demonstrate that any accumulation point of <span><math><msub><mrow><mo>{</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>ε</mi></mrow></msup><mo>}</mo></mrow><mrow><mn>0</mn><mo>&lt;</mo><mi>ε</mi><mo>⩽</mo><mn>1</mn></mrow></msub></math></span> can be regarded as a solution to the martingale problem or a weak solution of a distribution-dependent stochastic differential equation, which incorporates new drift and diffusion terms compared to the original equation. Our main contribution lies in employing two different methods to explicitly characterize the accumulation point. The diffusion matrices obtained through these two methods have different forms, however we assert their essential equivalence through a comparison.</p></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.4000,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Diffusion approximation for multi-scale McKean-Vlasov SDEs through different methods\",\"authors\":\"Wei Hong ,&nbsp;Shihu Li ,&nbsp;Xiaobin Sun\",\"doi\":\"10.1016/j.jde.2024.09.012\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>In this paper, our objective is to investigate the diffusion approximation for multi-scale McKean-Vlasov stochastic differential equations. More precisely, we first establish the tightness of the law of <span><math><msub><mrow><mo>{</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>ε</mi></mrow></msup><mo>}</mo></mrow><mrow><mn>0</mn><mo>&lt;</mo><mi>ε</mi><mo>⩽</mo><mn>1</mn></mrow></msub></math></span> in <span><math><mi>C</mi><mo>(</mo><mo>[</mo><mn>0</mn><mo>,</mo><mi>T</mi><mo>]</mo><mo>;</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></math></span>. Subsequently, we demonstrate that any accumulation point of <span><math><msub><mrow><mo>{</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>ε</mi></mrow></msup><mo>}</mo></mrow><mrow><mn>0</mn><mo>&lt;</mo><mi>ε</mi><mo>⩽</mo><mn>1</mn></mrow></msub></math></span> can be regarded as a solution to the martingale problem or a weak solution of a distribution-dependent stochastic differential equation, which incorporates new drift and diffusion terms compared to the original equation. Our main contribution lies in employing two different methods to explicitly characterize the accumulation point. The diffusion matrices obtained through these two methods have different forms, however we assert their essential equivalence through a comparison.</p></div>\",\"PeriodicalId\":15623,\"journal\":{\"name\":\"Journal of Differential Equations\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":2.4000,\"publicationDate\":\"2024-09-12\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Differential Equations\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0022039624005928\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Differential Equations","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022039624005928","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0

摘要

本文旨在研究多尺度麦金-弗拉索夫随机微分方程的扩散近似。更确切地说,我们首先建立了 C([0,T];Rn) 中 {Xε}0<ε⩽1 的严密性。随后,我们证明了{Xε}0<ε⩽1的任何累积点都可视为马丁格尔问题的解或依赖分布的随机微分方程的弱解,与原始方程相比,弱解包含了新的漂移和扩散项。我们的主要贡献在于采用了两种不同的方法来明确表征累积点。通过这两种方法得到的扩散矩阵具有不同的形式,但是我们通过比较来确定它们在本质上是等价的。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Diffusion approximation for multi-scale McKean-Vlasov SDEs through different methods

In this paper, our objective is to investigate the diffusion approximation for multi-scale McKean-Vlasov stochastic differential equations. More precisely, we first establish the tightness of the law of {Xε}0<ε1 in C([0,T];Rn). Subsequently, we demonstrate that any accumulation point of {Xε}0<ε1 can be regarded as a solution to the martingale problem or a weak solution of a distribution-dependent stochastic differential equation, which incorporates new drift and diffusion terms compared to the original equation. Our main contribution lies in employing two different methods to explicitly characterize the accumulation point. The diffusion matrices obtained through these two methods have different forms, however we assert their essential equivalence through a comparison.

求助全文
通过发布文献求助,成功后即可免费获取论文全文。 去求助
来源期刊
CiteScore
4.40
自引率
8.30%
发文量
543
审稿时长
9 months
期刊介绍: The Journal of Differential Equations is concerned with the theory and the application of differential equations. The articles published are addressed not only to mathematicians but also to those engineers, physicists, and other scientists for whom differential equations are valuable research tools. Research Areas Include: • Mathematical control theory • Ordinary differential equations • Partial differential equations • Stochastic differential equations • Topological dynamics • Related topics
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信