Fractional Backward Stochastic Partial Differential Equations with Applications to Stochastic Optimal Control of Partially Observed Systems driven by Lévy Processes
{"title":"Fractional Backward Stochastic Partial Differential Equations with Applications to Stochastic Optimal Control of Partially Observed Systems driven by Lévy Processes","authors":"Yuyang Ye, Yunzhang Li, Shanjian Tang","doi":"arxiv-2409.07052","DOIUrl":null,"url":null,"abstract":"In this paper, we study the Cauchy problem for backward stochastic partial\ndifferential equations (BSPDEs) involving fractional Laplacian operator.\nFirstly, by employing the martingale representation theorem and the fractional\nheat kernel, we construct an explicit form of the solution for fractional\nBSPDEs with space invariant coefficients, thereby demonstrating the existence\nand uniqueness of strong solution. Then utilizing the freezing coefficients\nmethod as well as the continuation method, we establish H\\\"older estimates and\nwell-posedness for general fractional BSPDEs with coefficients dependent on\nspace-time variables. As an application, we use the fractional adjoint BSPDEs\nto investigate stochastic optimal control of the partially observed systems\ndriven by $\\alpha$-stable L\\'evy processes.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"288 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Probability","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.07052","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper, we study the Cauchy problem for backward stochastic partial
differential equations (BSPDEs) involving fractional Laplacian operator.
Firstly, by employing the martingale representation theorem and the fractional
heat kernel, we construct an explicit form of the solution for fractional
BSPDEs with space invariant coefficients, thereby demonstrating the existence
and uniqueness of strong solution. Then utilizing the freezing coefficients
method as well as the continuation method, we establish H\"older estimates and
well-posedness for general fractional BSPDEs with coefficients dependent on
space-time variables. As an application, we use the fractional adjoint BSPDEs
to investigate stochastic optimal control of the partially observed systems
driven by $\alpha$-stable L\'evy processes.