{"title":"Tsallis value-at-risk: generalized entropic value-at-risk","authors":"Zhenfeng Zou, Zichao Xia, Taizhong Hu","doi":"10.1017/s0269964822000444","DOIUrl":null,"url":null,"abstract":"\n Motivated by Ahmadi-Javid (Journal of Optimization Theory Applications, 155(3), 2012, 1105–1123) and Ahmadi-Javid and Pichler (Mathematics and Financial Economics, 11, 2017, 527–550), the concept of Tsallis Value-at-Risk (TsVaR) based on Tsallis entropy is introduced in this paper. TsVaR corresponds to the tightest possible upper bound obtained from the Chernoff inequality for the Value-at-Risk. The main properties and analogous dual representation of TsVaR are investigated. These results partially generalize the Entropic Value-at-Risk by involving Tsallis entropies. Three spaces, called the primal, dual, and bidual Tsallis spaces, corresponding to TsVaR are fully studied. It is shown that these spaces equipped with the norm induced by TsVaR are Banach spaces. The Tsallis spaces are related to the \n \n \n $L^p$\n \n spaces, as well as specific Orlicz hearts and Orlicz spaces. Finally, we derive explicit formula for the dual TsVaR norm.","PeriodicalId":54582,"journal":{"name":"Probability in the Engineering and Informational Sciences","volume":"61 1","pages":""},"PeriodicalIF":0.7000,"publicationDate":"2022-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Probability in the Engineering and Informational Sciences","FirstCategoryId":"5","ListUrlMain":"https://doi.org/10.1017/s0269964822000444","RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"ENGINEERING, INDUSTRIAL","Score":null,"Total":0}
引用次数: 1
Abstract
Motivated by Ahmadi-Javid (Journal of Optimization Theory Applications, 155(3), 2012, 1105–1123) and Ahmadi-Javid and Pichler (Mathematics and Financial Economics, 11, 2017, 527–550), the concept of Tsallis Value-at-Risk (TsVaR) based on Tsallis entropy is introduced in this paper. TsVaR corresponds to the tightest possible upper bound obtained from the Chernoff inequality for the Value-at-Risk. The main properties and analogous dual representation of TsVaR are investigated. These results partially generalize the Entropic Value-at-Risk by involving Tsallis entropies. Three spaces, called the primal, dual, and bidual Tsallis spaces, corresponding to TsVaR are fully studied. It is shown that these spaces equipped with the norm induced by TsVaR are Banach spaces. The Tsallis spaces are related to the
$L^p$
spaces, as well as specific Orlicz hearts and Orlicz spaces. Finally, we derive explicit formula for the dual TsVaR norm.
本文在Ahmadi-Javid(优化理论应用，155(3)，2012,1105-1123)和Ahmadi-Javid and Pichler(数学与金融经济学，11,2017,527-550)的激励下，引入了基于Tsallis熵的Tsallis风险价值(TsVaR)概念。TsVaR对应于由Chernoff不等式得到的风险价值的最紧可能上界。研究了TsVaR的主要性质和类似对偶表示。这些结果通过涉及Tsallis熵部分地推广了风险熵。充分研究了与TsVaR相对应的三个空间，即原始、对偶和双Tsallis空间。结果表明，这些具有TsVaR诱导范数的空间是Banach空间。Tsallis空间与$L^p$空间以及特定的Orlicz心和Orlicz空间有关。最后，我们导出了对偶TsVaR范数的显式公式。
期刊介绍：
The primary focus of the journal is on stochastic modelling in the physical and engineering sciences, with particular emphasis on queueing theory, reliability theory, inventory theory, simulation, mathematical finance and probabilistic networks and graphs. Papers on analytic properties and related disciplines are also considered, as well as more general papers on applied and computational probability, if appropriate. Readers include academics working in statistics, operations research, computer science, engineering, management science and physical sciences as well as industrial practitioners engaged in telecommunications, computer science, financial engineering, operations research and management science.
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