Wasserstein perturbations of Markovian transition semigroups

IF 1.5 Q2 PHYSICS, MATHEMATICAL Annales de l Institut Henri Poincare D Pub Date : 2021-05-12 DOI:10.1214/22-aihp1270
Sven Fuhrmann, M. Kupper, M. Nendel
{"title":"Wasserstein perturbations of Markovian transition semigroups","authors":"Sven Fuhrmann, M. Kupper, M. Nendel","doi":"10.1214/22-aihp1270","DOIUrl":null,"url":null,"abstract":"In this paper, we deal with a class of time-homogeneous continuous-time Markov processes with transition probabilities bearing a nonparametric uncertainty. The uncertainty is modeled by considering perturbations of the transition probabilities within a proximity in Wasserstein distance. As a limit over progressively finer time periods, on which the level of uncertainty scales proportionally, we obtain a convex semigroup satisfying a nonlinear PDE in a viscosity sense. A remarkable observation is that, in standard situations, the nonlinear transition operators arising from nonparametric uncertainty coincide with the ones related to parametric drift uncertainty. On the level of the generator, the uncertainty is reflected as an additive perturbation in terms of a convex functional of first order derivatives. We additionally provide sensitivity bounds for the convex semigroup relative to the reference model. The results are illustrated with Wasserstein perturbations of L\\'evy processes, infinite-dimensional Ornstein-Uhlenbeck processes, geometric Brownian motions, and Koopman semigroups.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"37 1","pages":""},"PeriodicalIF":1.5000,"publicationDate":"2021-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"7","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annales de l Institut Henri Poincare D","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1214/22-aihp1270","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
引用次数: 7

Abstract

In this paper, we deal with a class of time-homogeneous continuous-time Markov processes with transition probabilities bearing a nonparametric uncertainty. The uncertainty is modeled by considering perturbations of the transition probabilities within a proximity in Wasserstein distance. As a limit over progressively finer time periods, on which the level of uncertainty scales proportionally, we obtain a convex semigroup satisfying a nonlinear PDE in a viscosity sense. A remarkable observation is that, in standard situations, the nonlinear transition operators arising from nonparametric uncertainty coincide with the ones related to parametric drift uncertainty. On the level of the generator, the uncertainty is reflected as an additive perturbation in terms of a convex functional of first order derivatives. We additionally provide sensitivity bounds for the convex semigroup relative to the reference model. The results are illustrated with Wasserstein perturbations of L\'evy processes, infinite-dimensional Ornstein-Uhlenbeck processes, geometric Brownian motions, and Koopman semigroups.
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
马尔可夫跃迁半群的Wasserstein摄动
本文研究了一类转移概率具有非参数不确定性的时间齐次连续马尔可夫过程。不确定性通过考虑Wasserstein距离内邻近的转移概率的扰动来建模。作为一个在越来越细的时间段上的极限,在这个时间段上,不确定性水平按比例缩放,我们得到了一个满足粘性意义上的非线性偏微分方程的凸半群。一个值得注意的观察是,在标准情况下,由非参数不确定性引起的非线性转移算子与与参数漂移不确定性有关的非线性转移算子重合。在生成器的层面上,不确定性反映为一阶导数的凸泛函形式的加性扰动。我们还提供了凸半群相对于参考模型的灵敏度界。结果用L′evy过程的Wasserstein摄动、无限维Ornstein-Uhlenbeck过程、几何布朗运动和Koopman半群来说明。
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
2.30
自引率
0.00%
发文量
16
期刊最新文献
A vertex model for supersymmetric LLT polynomials Duality of orthogonal and symplectic random tensor models Second order cumulants: Second order even elements and $R$-diagonal elements Fluctuations of dimer heights on contracting square-hexagon lattices Reflection of stochastic evolution equations in infinite dimensional domains
×
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