On Mean Field Stochastic Differential Equations Driven by $$G$$ -Brownian Motion with Averaging Principle

IF 0.8 Q2 MATHEMATICS Lobachevskii Journal of Mathematics Pub Date : 2024-07-19 DOI:10.1134/s1995080224600985
A. B. Touati, H. Boutabia, A. Redjil
{"title":"On Mean Field Stochastic Differential Equations Driven by $$G$$ -Brownian Motion with Averaging Principle","authors":"A. B. Touati, H. Boutabia, A. Redjil","doi":"10.1134/s1995080224600985","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>In a sublinear space <span>\\(\\left(\\Omega,\\mathcal{H},\\widehat{\\mathbb{E}}\\right)\\)</span>, we consider Mean Field stochastic differential equations (<span>\\(G\\)</span>-MFSDEs in short), called also <span>\\(G\\)</span>-McKean–Vlasov stochastic differential equations, which are SDEs where coefficients depend not only on the state of the unknown process but also on its law. We mean by law of a random variable <span>\\(X\\)</span> on <span>\\(\\left(\\Omega,\\mathcal{H},\\widehat{\\mathbb{E}}\\right)\\)</span>, the set <span>\\(\\left\\{P_{X}:P\\in\\mathcal{P}\\right\\}\\)</span>, where <span>\\(P_{X}\\)</span> is the law of <span>\\(X\\)</span> with respect to <span>\\(P\\)</span> and <span>\\(\\mathcal{P}\\)</span> is the family of probabilities associated to the sublinear expectation <span>\\(\\widehat{\\mathbb{E}}\\)</span>. In this paper, we study the existence and uniqueness of the solution of <span>\\(G\\)</span>-MFSDE by using the fixed point theorem. To this end, we introduce a new type Kantorovich metric between subsets of laws and adapted Lipchitz and linear growth conditions. Furthermore, we prove the validity of the averaging principle and obtain convergence theorem where the solution of the averaged <span>\\(G\\)</span>-MFSDE converges to that of the standard one in the mean square sense.</p>","PeriodicalId":46135,"journal":{"name":"Lobachevskii Journal of Mathematics","volume":"64 1","pages":""},"PeriodicalIF":0.8000,"publicationDate":"2024-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Lobachevskii Journal of Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1134/s1995080224600985","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0

Abstract

In a sublinear space \(\left(\Omega,\mathcal{H},\widehat{\mathbb{E}}\right)\), we consider Mean Field stochastic differential equations (\(G\)-MFSDEs in short), called also \(G\)-McKean–Vlasov stochastic differential equations, which are SDEs where coefficients depend not only on the state of the unknown process but also on its law. We mean by law of a random variable \(X\) on \(\left(\Omega,\mathcal{H},\widehat{\mathbb{E}}\right)\), the set \(\left\{P_{X}:P\in\mathcal{P}\right\}\), where \(P_{X}\) is the law of \(X\) with respect to \(P\) and \(\mathcal{P}\) is the family of probabilities associated to the sublinear expectation \(\widehat{\mathbb{E}}\). In this paper, we study the existence and uniqueness of the solution of \(G\)-MFSDE by using the fixed point theorem. To this end, we introduce a new type Kantorovich metric between subsets of laws and adapted Lipchitz and linear growth conditions. Furthermore, we prove the validity of the averaging principle and obtain convergence theorem where the solution of the averaged \(G\)-MFSDE converges to that of the standard one in the mean square sense.

查看原文
分享 分享
微信好友 朋友圈 QQ好友 复制链接
本刊更多论文
带平均原则的$$G$$布朗运动驱动的平均场随机微分方程
Abstract In a sublinear space \(\left(\Omega,\mathcal{H},\widehat\mathbb{E}}right)\),we consider Mean Field stochastic differential equations (简称\(G\)-MFSDEs), called also \(G\)-McKean-Vlasov stochastic differential equations, which are SDEs where cofficients depend on not only the state of unknown process but also on its law.我们所说的随机变量(X)的规律是指(left(\Omega,\mathcal{H},\widehat{mathbb{E}}\right))上的集合(\left\{P_{X}:其中,\(P_{X}\)是\(X)关于\(P\)的规律,而\(\mathcal{P}\)是与亚线性期望\(\widehat\{mathbb{E}}\)相关的概率族。本文利用定点定理研究了 \(G\)-MFSDE 解的存在性和唯一性。为此,我们在定律子集之间引入了一种新型康托洛维奇度量,并调整了李普希兹条件和线性增长条件。此外,我们证明了平均原理的有效性,并得到了收敛定理,即平均 \(G\)-MFSDE 的解在均方意义上收敛于标准解。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 去求助
来源期刊
CiteScore
1.50
自引率
42.90%
发文量
127
期刊介绍: Lobachevskii Journal of Mathematics is an international peer reviewed journal published in collaboration with the Russian Academy of Sciences and Kazan Federal University. The journal covers mathematical topics associated with the name of famous Russian mathematician Nikolai Lobachevsky (Lobachevskii). The journal publishes research articles on geometry and topology, algebra, complex analysis, functional analysis, differential equations and mathematical physics, probability theory and stochastic processes, computational mathematics, mathematical modeling, numerical methods and program complexes, computer science, optimal control, and theory of algorithms as well as applied mathematics. The journal welcomes manuscripts from all countries in the English language.
期刊最新文献
Oscillations of Nanofilms in a Fluid Pressure Diffusion Waves in a Porous Medium Saturated by Three Phase Fluid Effect of a Rigid Cone Inserted in a Tube on Resonant Gas Oscillations Taylor Nearly Columnar Vortices in the Couette–Taylor System: Transition to Turbulence From Texts to Knowledge Graph in the Semantic Library LibMeta
×
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
现在去查看 取消
×
提示
确定
0
微信
客服QQ
Book学术公众号 扫码关注我们
反馈
×
意见反馈
请填写您的意见或建议
请填写您的手机或邮箱
已复制链接
已复制链接
快去分享给好友吧!
我知道了
×
扫码分享
扫码分享
Book学术官方微信
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术
文献互助 智能选刊 最新文献 互助须知 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 Book学术 All rights reserved.
ghs 京公网安备 11010802042870号 京ICP备2023020795号-1