{"title":"Discrete stopping times in the lattice of continuous functions","authors":"Achintya Raya Polavarapu","doi":"10.1007/s11117-024-01044-5","DOIUrl":null,"url":null,"abstract":"<p>A functional calculus for an order complete vector lattice <span>\\({\\mathcal {E}}\\)</span> was developed by Grobler (Indag Math (NS) 25(2):275–295, 2014) using the Daniell integral. We show that if one represents the universal completion of <span>\\({\\mathcal {E}}\\)</span> as <span>\\(C^\\infty (K)\\)</span>, where <i>K</i> is an extremally disconnected compact Hausdorff topological space, then the Daniell functional calculus for continuous functions is exactly the pointwise composition of functions in <span>\\(C^\\infty (K)\\)</span>. This representation allows an easy deduction of the various properties of the functional calculus. Afterwards, we study discrete stopping times and stopped processes in <span>\\(C^\\infty (K)\\)</span>. We obtain a representation that is analogous to what is expected in probability theory.\n</p>","PeriodicalId":54596,"journal":{"name":"Positivity","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2024-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Positivity","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s11117-024-01044-5","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
A functional calculus for an order complete vector lattice \({\mathcal {E}}\) was developed by Grobler (Indag Math (NS) 25(2):275–295, 2014) using the Daniell integral. We show that if one represents the universal completion of \({\mathcal {E}}\) as \(C^\infty (K)\), where K is an extremally disconnected compact Hausdorff topological space, then the Daniell functional calculus for continuous functions is exactly the pointwise composition of functions in \(C^\infty (K)\). This representation allows an easy deduction of the various properties of the functional calculus. Afterwards, we study discrete stopping times and stopped processes in \(C^\infty (K)\). We obtain a representation that is analogous to what is expected in probability theory.
期刊介绍：
The purpose of Positivity is to provide an outlet for high quality original research in all areas of analysis and its applications to other disciplines having a clear and substantive link to the general theme of positivity. Specifically, articles that illustrate applications of positivity to other disciplines - including but not limited to - economics, engineering, life sciences, physics and statistical decision theory are welcome.
The scope of Positivity is to publish original papers in all areas of mathematics and its applications that are influenced by positivity concepts.
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