{"title":"具有多元点过程和右上半连续障碍的反射BSDE的存在唯一性","authors":"Baadi, Brahim, Marzougue, Mohamed","doi":"10.1515/rose-2023-2019","DOIUrl":null,"url":null,"abstract":"Abstract In a noise driven by a multivariate point process μ with predictable compensator ν, we prove existence and uniqueness of the reflected backward stochastic differential equation’s solution with a lower obstacle <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:msub> <m:mi>ξ</m:mi> <m:mi>t</m:mi> </m:msub> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mrow> <m:mi>t</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy=\"false\">[</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mi>T</m:mi> <m:mo stretchy=\"false\">]</m:mo> </m:mrow> </m:mrow> </m:msub> </m:math> {(\\xi_{t})_{t\\in[0,T]}} which is assumed to be a right upper-semicontinuous, but not necessarily right-continuous process, and a Lipschitz driver f . The result is established by using the Mertens decomposition of optional strong (but not necessarily right continuous) super-martingales, an appropriate generalization of Itô’s formula due to Gal’chouk and Lenglart and some tools from optimal stopping theory. A comparison theorem for this type of equations is given.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":"48 6","pages":"0"},"PeriodicalIF":0.3000,"publicationDate":"2023-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Existence and uniqueness for reflected BSDE with multivariate point process and right upper semicontinuous obstacle\",\"authors\":\"Baadi, Brahim, Marzougue, Mohamed\",\"doi\":\"10.1515/rose-2023-2019\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Abstract In a noise driven by a multivariate point process μ with predictable compensator ν, we prove existence and uniqueness of the reflected backward stochastic differential equation’s solution with a lower obstacle <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:msub> <m:mrow> <m:mo stretchy=\\\"false\\\">(</m:mo> <m:msub> <m:mi>ξ</m:mi> <m:mi>t</m:mi> </m:msub> <m:mo stretchy=\\\"false\\\">)</m:mo> </m:mrow> <m:mrow> <m:mi>t</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy=\\\"false\\\">[</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mi>T</m:mi> <m:mo stretchy=\\\"false\\\">]</m:mo> </m:mrow> </m:mrow> </m:msub> </m:math> {(\\\\xi_{t})_{t\\\\in[0,T]}} which is assumed to be a right upper-semicontinuous, but not necessarily right-continuous process, and a Lipschitz driver f . The result is established by using the Mertens decomposition of optional strong (but not necessarily right continuous) super-martingales, an appropriate generalization of Itô’s formula due to Gal’chouk and Lenglart and some tools from optimal stopping theory. A comparison theorem for this type of equations is given.\",\"PeriodicalId\":43421,\"journal\":{\"name\":\"Random Operators and Stochastic Equations\",\"volume\":\"48 6\",\"pages\":\"0\"},\"PeriodicalIF\":0.3000,\"publicationDate\":\"2023-10-27\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Random Operators and Stochastic Equations\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1515/rose-2023-2019\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"STATISTICS & PROBABILITY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Random Operators and Stochastic Equations","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1515/rose-2023-2019","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
引用次数: 0
摘要
摘要在具有可预测补偿器ν的多元点过程μ驱动的噪声中,证明了具有下障碍(ξ t) t∈[0,t] {(\xi_{t})_{t\ In [0, t]}}的反射后向随机微分方程解的存在唯一性,该方程被假设为右上半连续过程,但不一定是右连续过程,并具有Lipschitz驱动器f。利用可选强(但不一定是正确连续)超鞅的Mertens分解、Gal ' chouk和Lenglart对Itô公式的适当推广以及最优停止理论中的一些工具,建立了该结果。给出了这类方程的一个比较定理。
Existence and uniqueness for reflected BSDE with multivariate point process and right upper semicontinuous obstacle
Abstract In a noise driven by a multivariate point process μ with predictable compensator ν, we prove existence and uniqueness of the reflected backward stochastic differential equation’s solution with a lower obstacle (ξt)t∈[0,T] {(\xi_{t})_{t\in[0,T]}} which is assumed to be a right upper-semicontinuous, but not necessarily right-continuous process, and a Lipschitz driver f . The result is established by using the Mertens decomposition of optional strong (but not necessarily right continuous) super-martingales, an appropriate generalization of Itô’s formula due to Gal’chouk and Lenglart and some tools from optimal stopping theory. A comparison theorem for this type of equations is given.