Pub Date : 2019-12-01DOI: 10.1186/S41546-019-0037-3
Renzhi Qiu, Shanjian Tang
{"title":"The Cauchy problem of Backward Stochastic Super-Parabolic Equations with Quadratic Growth","authors":"Renzhi Qiu, Shanjian Tang","doi":"10.1186/S41546-019-0037-3","DOIUrl":"https://doi.org/10.1186/S41546-019-0037-3","url":null,"abstract":"","PeriodicalId":42330,"journal":{"name":"Probability Uncertainty and Quantitative Risk","volume":"14 1","pages":"1-29"},"PeriodicalIF":1.5,"publicationDate":"2019-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84540838","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper establishes an existence and uniqueness result for the adapted solution of a general time interval multidimensional backward stochastic differential equation (BSDE), where the generator begin{document}$ g $end{document} satisfies a weak stochastic-monotonicity condition and a general growth condition in the state variable begin{document}$ y $end{document}, and a stochastic-Lipschitz condition in the state variable begin{document}$ z $end{document}. This unifies and strengthens some known works. In order to prove this result, we develop some ideas and techniques employed in Xiao and Fan [25] and Liu et al. [15]. In particular, we put forward and prove a stochastic Gronwall-type inequality and a stochastic Bihari-type inequality, which generalize the classical ones and may be useful in other applications. The martingale representation theorem, Itô’s formula, and the BMO martingale tool are used to prove these two inequalities.
This paper establishes an existence and uniqueness result for the adapted solution of a general time interval multidimensional backward stochastic differential equation (BSDE), where the generator begin{document}$ g $end{document} satisfies a weak stochastic-monotonicity condition and a general growth condition in the state variable begin{document}$ y $end{document} , and a stochastic-Lipschitz condition in the state variable begin{document}$ z $end{document} . This unifies and strengthens some known works. In order to prove this result, we develop some ideas and techniques employed in Xiao and Fan [25] and Liu et al. [15]. In particular, we put forward and prove a stochastic Gronwall-type inequality and a stochastic Bihari-type inequality, which generalize the classical ones and may be useful in other applications. The martingale representation theorem, Itô’s formula, and the BMO martingale tool are used to prove these two inequalities.
{"title":"General time interval multidimensional BSDEs with generators satisfying a weak stochastic-monotonicity condition","authors":"Tingting Li, Ziheng Xu, Shengjun Fan","doi":"10.3934/puqr.2021015","DOIUrl":"https://doi.org/10.3934/puqr.2021015","url":null,"abstract":"<p style='text-indent:20px;'>This paper establishes an existence and uniqueness result for the adapted solution of a general time interval multidimensional backward stochastic differential equation (BSDE), where the generator <inline-formula> <tex-math id=\"M1\">begin{document}$ g $end{document}</tex-math> </inline-formula> satisfies a weak stochastic-monotonicity condition and a general growth condition in the state variable <inline-formula> <tex-math id=\"M2\">begin{document}$ y $end{document}</tex-math> </inline-formula>, and a stochastic-Lipschitz condition in the state variable <inline-formula> <tex-math id=\"M3\">begin{document}$ z $end{document}</tex-math> </inline-formula>. This unifies and strengthens some known works. In order to prove this result, we develop some ideas and techniques employed in Xiao and Fan [<xref ref-type=\"bibr\" rid=\"b25\">25</xref>] and Liu et al. [<xref ref-type=\"bibr\" rid=\"b15\">15</xref>]. In particular, we put forward and prove a stochastic Gronwall-type inequality and a stochastic Bihari-type inequality, which generalize the classical ones and may be useful in other applications. The martingale representation theorem, Itô’s formula, and the BMO martingale tool are used to prove these two inequalities. </p>","PeriodicalId":42330,"journal":{"name":"Probability Uncertainty and Quantitative Risk","volume":"62 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2019-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78552676","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
An expectile can be considered a generalization of a quantile. While expected shortfall is a quantile-based risk measure, we study its counterpart—the expectile-based expected shortfall—where expectile takes the place of a quantile. We provide its dual representation in terms of a Bochner integral. Among other properties, we show that it is bounded from below in terms of the convex combination of expected shortfalls, and also from above by the smallest law invariant, coherent, and comonotonic risk measures, for which we give the explicit formulation of the corresponding distortion function. As a benchmark to the industry standard expected shortfall, we further provide its comparative asymptotic behavior in terms of extreme value distributions. Based on these results, we finally explicitly compute the expectile-based expected shortfall for selected classes of distributions.
{"title":"Dual representation of expectile-based expected shortfall and its properties","authors":"Samuel Drapeau, Mekonnen Tadese","doi":"10.3934/puqr.2021005","DOIUrl":"https://doi.org/10.3934/puqr.2021005","url":null,"abstract":"An expectile can be considered a generalization of a quantile. While expected shortfall is a quantile-based risk measure, we study its counterpart—the expectile-based expected shortfall—where expectile takes the place of a quantile. We provide its dual representation in terms of a Bochner integral. Among other properties, we show that it is bounded from below in terms of the convex combination of expected shortfalls, and also from above by the smallest law invariant, coherent, and comonotonic risk measures, for which we give the explicit formulation of the corresponding distortion function. As a benchmark to the industry standard expected shortfall, we further provide its comparative asymptotic behavior in terms of extreme value distributions. Based on these results, we finally explicitly compute the expectile-based expected shortfall for selected classes of distributions.","PeriodicalId":42330,"journal":{"name":"Probability Uncertainty and Quantitative Risk","volume":"29 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2019-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90609911","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-08-26DOI: 10.1186/s41546-019-0041-7
Uncertainty and Quantitative Risk Probability
{"title":"Publisher Correction to: Probability, uncertainty and quantitative risk, volume 4","authors":"Uncertainty and Quantitative Risk Probability","doi":"10.1186/s41546-019-0041-7","DOIUrl":"https://doi.org/10.1186/s41546-019-0041-7","url":null,"abstract":"","PeriodicalId":42330,"journal":{"name":"Probability Uncertainty and Quantitative Risk","volume":"87 1","pages":"1"},"PeriodicalIF":1.5,"publicationDate":"2019-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83437438","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Peng, S. [6] proved the law of large numbers under a sublinear expectation. In this paper, we give its error estimates by Stein’s method.
Peng, S.[6]证明了次线性期望下的大数定律。本文用Stein方法给出了其误差估计。
{"title":"Stein’s method for the law of large numbers under sublinear expectations","authors":"Yongsheng Song","doi":"10.3934/puqr.2021010","DOIUrl":"https://doi.org/10.3934/puqr.2021010","url":null,"abstract":"<p style='text-indent:20px;'>Peng, S. [<xref ref-type=\"bibr\" rid=\"b6\">6</xref>] proved the law of large numbers under a sublinear expectation. In this paper, we give its error estimates by Stein’s method. </p>","PeriodicalId":42330,"journal":{"name":"Probability Uncertainty and Quantitative Risk","volume":"1 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2019-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76956340","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-12-27DOI: 10.1186/s41546-020-00048-9
Ludovic Tangpi
{"title":"Efficient hedging under ambiguity in continuous time","authors":"Ludovic Tangpi","doi":"10.1186/s41546-020-00048-9","DOIUrl":"https://doi.org/10.1186/s41546-020-00048-9","url":null,"abstract":"","PeriodicalId":42330,"journal":{"name":"Probability Uncertainty and Quantitative Risk","volume":"125 1","pages":"1-19"},"PeriodicalIF":1.5,"publicationDate":"2018-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75929683","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Harnack inequality and gradient estimate for functional G-SDEs with degenerate noise","authors":"Xing Huang, Fen-Fen Yang","doi":"10.3934/puqr.2022008","DOIUrl":"https://doi.org/10.3934/puqr.2022008","url":null,"abstract":"","PeriodicalId":42330,"journal":{"name":"Probability Uncertainty and Quantitative Risk","volume":"28 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2018-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83126279","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-11-03DOI: 10.1186/s41546-020-00047-w
Jiequn Han, Jihao Long
{"title":"Convergence of the deep BSDE method for coupled FBSDEs","authors":"Jiequn Han, Jihao Long","doi":"10.1186/s41546-020-00047-w","DOIUrl":"https://doi.org/10.1186/s41546-020-00047-w","url":null,"abstract":"","PeriodicalId":42330,"journal":{"name":"Probability Uncertainty and Quantitative Risk","volume":"31 1","pages":"1-33"},"PeriodicalIF":1.5,"publicationDate":"2018-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73957029","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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