Abstract We study a preferred equity infusion government program set to mitigate interbank contagion. Financial institutions are prone to insolvency risk channeled through the network of interbank debt and to funding liquidity risk. The government seeks to maximize, under budget constraints, the total net worth of the financial system or, equivalently, to minimize the dead-weight losses induced by bank runs. The government is assumed to have complete information on interbank debt. The problem of quantifying the optimal amount of infusions can be expressed as a convex combinatorial optimization problem, tractable when the set of banks eligible for intervention (core banks) is sufficiently, yet realistically, small. We find that no bank has an incentive to withdraw from the program, when the preferred dividend rate paid to the government is equal to the government's outside return on the intervention budget. On the other hand, it may be optimal for the government to make infusions in a strict subset of core banks.
{"title":"Optimal control of interbank contagion under complete information","authors":"Andreea Minca, A. Sulem","doi":"10.1515/strm-2013-1165","DOIUrl":"https://doi.org/10.1515/strm-2013-1165","url":null,"abstract":"Abstract We study a preferred equity infusion government program set to mitigate interbank contagion. Financial institutions are prone to insolvency risk channeled through the network of interbank debt and to funding liquidity risk. The government seeks to maximize, under budget constraints, the total net worth of the financial system or, equivalently, to minimize the dead-weight losses induced by bank runs. The government is assumed to have complete information on interbank debt. The problem of quantifying the optimal amount of infusions can be expressed as a convex combinatorial optimization problem, tractable when the set of banks eligible for intervention (core banks) is sufficiently, yet realistically, small. We find that no bank has an incentive to withdraw from the program, when the preferred dividend rate paid to the government is equal to the government's outside return on the intervention budget. On the other hand, it may be optimal for the government to make infusions in a strict subset of core banks.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2013-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/strm-2013-1165","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67313391","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The demand for an accurate financial risk management involving larger numbers of assets is strong not only in view of the financial crisis of 2007–2009. Especially dependencies among assets have not been captured adequately. While standard multivariate copulas have added some flexibility, this flexibility is insufficient in higher dimensional applications. Vine copulas can fill this gap by benefiting from the rich class of existing bivariate parametric copula families. Exploiting this in combination with GARCH models for the margins, we develop a regular vine copula based factor model for asset returns, the Regular Vine Market Sector model, which is motivated by the classical CAPM and shown to be superior to the CAVA model proposed by Heinen and Valdesogo (2009). The model can also be used to separate the systematic and idiosyncratic risk of specific stocks, and we explicitly discuss how vine copula models can be employed for active and passive portfolio management. In particular, Value-at-Risk forecasting and asset allocation are treated in detail. All developed models and methods are used to analyze the Euro Stoxx 50 index, a major market indicator for the Eurozone. Relevant benchmark models such as the popular DCC model and the common Student's t copula are taken into account.
摘要2007-2009年金融危机爆发后，对涉及更大规模资产的准确金融风险管理的需求日益强烈。特别是资产之间的依赖关系没有被充分捕获。虽然标准的多元copuls增加了一些灵活性，但这种灵活性在高维应用中是不够的。Vine copula可以利用现有的丰富的二元参数copula族来填补这一空白。利用这一理论与GARCH边际模型相结合，我们开发了一个基于常规vine copula的资产回报因子模型，即常规vine市场部门模型，该模型由经典CAPM驱动，并被证明优于Heinen和Valdesogo(2009)提出的CAVA模型。该模型还可以用于分离特定股票的系统风险和特质风险，并明确讨论了如何将藤联结模型用于主动和被动投资组合管理。特别是，在风险价值预测和资产配置的详细处理。所有开发的模型和方法都用于分析欧洲斯托克50指数，这是欧元区的主要市场指标。考虑了相关的基准模型，如流行的DCC模型和常见的Student's t copula。
{"title":"Risk management with high-dimensional vine copulas: An analysis of the Euro Stoxx 50","authors":"E. Brechmann, C. Czado","doi":"10.1524/strm.2013.2002","DOIUrl":"https://doi.org/10.1524/strm.2013.2002","url":null,"abstract":"Abstract The demand for an accurate financial risk management involving larger numbers of assets is strong not only in view of the financial crisis of 2007–2009. Especially dependencies among assets have not been captured adequately. While standard multivariate copulas have added some flexibility, this flexibility is insufficient in higher dimensional applications. Vine copulas can fill this gap by benefiting from the rich class of existing bivariate parametric copula families. Exploiting this in combination with GARCH models for the margins, we develop a regular vine copula based factor model for asset returns, the Regular Vine Market Sector model, which is motivated by the classical CAPM and shown to be superior to the CAVA model proposed by Heinen and Valdesogo (2009). The model can also be used to separate the systematic and idiosyncratic risk of specific stocks, and we explicitly discuss how vine copula models can be employed for active and passive portfolio management. In particular, Value-at-Risk forecasting and asset allocation are treated in detail. All developed models and methods are used to analyze the Euro Stoxx 50 index, a major market indicator for the Eurozone. Relevant benchmark models such as the popular DCC model and the common Student's t copula are taken into account.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2013-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1524/strm.2013.2002","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66893105","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract There is an increasing demand for models of multivariate time-series with time-varying and non-Gaussian dependencies. The available models suffer from the curse of dimensionality or from restrictive assumptions on the parameters and distributions. A promising class of models is that of hierarchical Archimedean copulae (HAC), which allows for non-exchangeable and non-Gaussian dependency structures with a small number of parameters. In this paper we develop a novel adaptive estimation technique of the parameters and of the structure of HAC for time-series. The approach relies on a local change-point detection procedure and a locally constant HAC approximation. Typical applications are in the financial area but also recently in the spatial analysis of weather parameters. We analyse the time varying dependency structure of stock indices and exchange rates. Both examples reveal periods with constant and turmoil dependencies. The economic significance of the suggested modelling is evaluated using the Value-at-Risk of a portfolio.
{"title":"Dynamic structured copula models","authors":"W. Härdle, Ostap Okhrin, Yarema Okhrin","doi":"10.1524/strm.2013.2004","DOIUrl":"https://doi.org/10.1524/strm.2013.2004","url":null,"abstract":"Abstract There is an increasing demand for models of multivariate time-series with time-varying and non-Gaussian dependencies. The available models suffer from the curse of dimensionality or from restrictive assumptions on the parameters and distributions. A promising class of models is that of hierarchical Archimedean copulae (HAC), which allows for non-exchangeable and non-Gaussian dependency structures with a small number of parameters. In this paper we develop a novel adaptive estimation technique of the parameters and of the structure of HAC for time-series. The approach relies on a local change-point detection procedure and a locally constant HAC approximation. Typical applications are in the financial area but also recently in the spatial analysis of weather parameters. We analyse the time varying dependency structure of stock indices and exchange rates. Both examples reveal periods with constant and turmoil dependencies. The economic significance of the suggested modelling is evaluated using the Value-at-Risk of a portfolio.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2013-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1524/strm.2013.2004","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66893294","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this paper we study asymptotically stable risk assessments (or equivalently risk measures) which have the property that an unacceptable position cannot become acceptable by adding a huge cash-flow far in the future. Under an additional continuity assumption, these risk assessments are exactly those which have a robust representation in terms of test probabilities that are supported on a finite time interval. For time-consistent risk assessments we give conditions on their generators which guarantee asymptotic stability.
{"title":"Asymptotically stable dynamic risk assessments","authors":"Karl-Theodor Eisele, M. Kupper","doi":"10.1515/strm-2012-1146","DOIUrl":"https://doi.org/10.1515/strm-2012-1146","url":null,"abstract":"Abstract In this paper we study asymptotically stable risk assessments (or equivalently risk measures) which have the property that an unacceptable position cannot become acceptable by adding a huge cash-flow far in the future. Under an additional continuity assumption, these risk assessments are exactly those which have a robust representation in terms of test probabilities that are supported on a finite time interval. For time-consistent risk assessments we give conditions on their generators which guarantee asymptotic stability.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2013-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/strm-2012-1146","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67312874","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Copulae became an extremely popular tool in different areas of research. Since the first applications in risk management in the late 90th, they attracted many other quantitatively oriented sciences like biostatistics, hydrology and finance. The main reason originates in the Sklar (1959) theorem, which allows for separation of the marginal distributions from the dependency structure between the random variables. This editorial is organized as follows. In the first section we define the copulae and state the Sklar theorem. Some literature suggestions are given in the second section. The last section presents the content of this special issue.
{"title":"Editorial to the special issue on Copulae of Statistics & Risk Modeling","authors":"Ostap Okhrin","doi":"10.1524/strm.2013.9014","DOIUrl":"https://doi.org/10.1524/strm.2013.9014","url":null,"abstract":"Abstract Copulae became an extremely popular tool in different areas of research. Since the first applications in risk management in the late 90th, they attracted many other quantitatively oriented sciences like biostatistics, hydrology and finance. The main reason originates in the Sklar (1959) theorem, which allows for separation of the marginal distributions from the dependency structure between the random variables. This editorial is organized as follows. In the first section we define the copulae and state the Sklar theorem. Some literature suggestions are given in the second section. The last section presents the content of this special issue.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2013-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1524/strm.2013.9014","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66893409","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We present a list of challenges one faces when given the task of modeling dependence between stochastic objects, with a special focus on financial applications. Our aim is to draw the readers' attention to common (and not so common) pitfalls and fallacies, and we particularly address readers who are new to dependence modeling. The presented list of challenges is clearly not complete, but it gives a flavor of how difficult and subtle the task of dependence modeling can be. Moreover, the readers shall get some intuition about what challenges are structural and cannot be overcome, and what challenges allow for a better solution than common practice might suggest.
{"title":"What makes dependence modeling challenging? Pitfalls and ways to circumvent them","authors":"Jan-Frederik Mai, M. Scherer","doi":"10.1524/strm.2013.2001","DOIUrl":"https://doi.org/10.1524/strm.2013.2001","url":null,"abstract":"Abstract We present a list of challenges one faces when given the task of modeling dependence between stochastic objects, with a special focus on financial applications. Our aim is to draw the readers' attention to common (and not so common) pitfalls and fallacies, and we particularly address readers who are new to dependence modeling. The presented list of challenges is clearly not complete, but it gives a flavor of how difficult and subtle the task of dependence modeling can be. Moreover, the readers shall get some intuition about what challenges are structural and cannot be overcome, and what challenges allow for a better solution than common practice might suggest.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2013-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1524/strm.2013.2001","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66893075","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Copulas are widely used for modeling the dependence structure of multivariate data. Many methods for estimating the copula density functions are investigated. In this paper, we study the asymptotic properties of the Bernstein estimator for unbounded copula density functions. We show that the estimator converges to infinity at the corner and we establish its relative convergence when the copula density is unbounded. Also, we provide the uniform strong consistency of the estimator on every compact in the interior region. We investigate the finite sample performance of the estimator via an extensive simulation study and we compare the Bernstein copula density estimator with other nonparametric methods. Finally, we consider an empirical application where the asymmetric dependence between international equity markets (US, Canada, UK, and France) is examined.
{"title":"Bernstein estimator for unbounded copula densities","authors":"T. Bouezmarni, El Ghouch, A. Taamouti","doi":"10.1524/strm.2013.2003","DOIUrl":"https://doi.org/10.1524/strm.2013.2003","url":null,"abstract":"Abstract Copulas are widely used for modeling the dependence structure of multivariate data. Many methods for estimating the copula density functions are investigated. In this paper, we study the asymptotic properties of the Bernstein estimator for unbounded copula density functions. We show that the estimator converges to infinity at the corner and we establish its relative convergence when the copula density is unbounded. Also, we provide the uniform strong consistency of the estimator on every compact in the interior region. We investigate the finite sample performance of the estimator via an extensive simulation study and we compare the Bernstein copula density estimator with other nonparametric methods. Finally, we consider an empirical application where the asymmetric dependence between international equity markets (US, Canada, UK, and France) is examined.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2013-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1524/strm.2013.2003","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66893214","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We study the impact of central clearing of over-the-counter (OTC) transactions on counterparty exposures in a market with OTC transactions across several asset classes with heterogeneous characteristics. The impact of introducing a central counterparty (CCP) on expected interdealer exposure is determined by the tradeoff between multilateral netting across dealers on one hand and bilateral netting across asset classes on the other hand. We find this tradeoff to be sensitive to assumptions on heterogeneity of asset classes in terms of `riskyness' of the asset class as well as correlation of exposures across asset classes. In particular, while an analysis assuming independent, homogeneous exposures suggests that central clearing is efficient only if one has an unrealistically high number of participants, the opposite conclusion is reached if differences in riskyness and correlation across asset classes are realistically taken into account. We argue that empirically plausible specifications of model parameters lead to the conclusion that central clearing does reduce interdealer exposures: the gain from multilateral netting in a CCP overweighs the loss of netting across asset classes in bilateral netting agreements. When a CCP exists for interest rate derivatives, adding a CCP for credit derivatives is shown to decrease overall exposures. These findings are shown to be robust to the statistical assumptions of the model as well as the choice of risk measure used to quantify exposures.
{"title":"Central clearing of OTC derivatives: Bilateral vs multilateral netting","authors":"R. Cont, Thomas Kokholm","doi":"10.1515/strm-2013-1161","DOIUrl":"https://doi.org/10.1515/strm-2013-1161","url":null,"abstract":"Abstract We study the impact of central clearing of over-the-counter (OTC) transactions on counterparty exposures in a market with OTC transactions across several asset classes with heterogeneous characteristics. The impact of introducing a central counterparty (CCP) on expected interdealer exposure is determined by the tradeoff between multilateral netting across dealers on one hand and bilateral netting across asset classes on the other hand. We find this tradeoff to be sensitive to assumptions on heterogeneity of asset classes in terms of `riskyness' of the asset class as well as correlation of exposures across asset classes. In particular, while an analysis assuming independent, homogeneous exposures suggests that central clearing is efficient only if one has an unrealistically high number of participants, the opposite conclusion is reached if differences in riskyness and correlation across asset classes are realistically taken into account. We argue that empirically plausible specifications of model parameters lead to the conclusion that central clearing does reduce interdealer exposures: the gain from multilateral netting in a CCP overweighs the loss of netting across asset classes in bilateral netting agreements. When a CCP exists for interest rate derivatives, adding a CCP for credit derivatives is shown to decrease overall exposures. These findings are shown to be robust to the statistical assumptions of the model as well as the choice of risk measure used to quantify exposures.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2012-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/strm-2013-1161","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67313249","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract This paper is dedicated to the consistency of systemic risk measures with respect to stochastic dependence. It compares two alternative notions of Conditional Value-at-Risk (CoVaR) available in the current literature. These notions are both based on the conditional distribution of a random variable Y given a stress event for a random variable X , but they use different types of stress events. We derive representations of these alternative CoVaR notions in terms of copulas, study their general dependence consistency and compare their performance in several stochastic models. Our central finding is that conditioning on X ≥ VaR α ( X ) gives a much better response to dependence between X and Y than conditioning on X = VaR α ( X ). We prove general results that relate the dependence consistency of CoVaR using conditioning on X ≥ VaR α ( X ) to well established results on concordance ordering of multivariate distributions or their copulas. These results also apply to some other systemic risk measures, such as the Marginal Expected Shortfall (MES) and the Systemic Impact Index (SII). We provide counterexamples showing that CoVaR based on the stress event X = VaR α ( X ) is not dependence consistent. In particular, if ( X , Y ) is bivariate normal, then CoVaR based on X = VaR α ( X ) is not an increasing function of the correlation parameter. Similar issues arise in the bivariate t model and in the model with t margins and a Gumbel copula. In all these cases, CoVaR based on X ≥ VaR α ( X ) is an increasing function of the dependence parameter.
摘要研究随机依赖下系统风险测度的一致性问题。它比较了当前文献中可用的条件风险价值(CoVaR)的两种替代概念。这些概念都是基于给定随机变量X的压力事件的随机变量Y的条件分布，但它们使用不同类型的压力事件。我们推导了这些备选CoVaR概念在copula中的表示，研究了它们的一般依赖一致性，并比较了它们在几个随机模型中的表现。我们的中心发现是，条件X≥VaR α (X)比条件X = VaR α (X)对X和Y之间的依赖性有更好的反应。我们证明了在X≥VaR α (X)条件下CoVaR的依赖一致性与多元分布或它们的联结的一致性排序的一般结果。这些结果也适用于其他一些系统性风险指标，如边际预期缺口(MES)和系统影响指数(SII)。我们提供了反例，表明基于应力事件X = VaR α (X)的CoVaR不是依赖一致的。特别是，如果(X, Y)是二元正态，则基于X = VaR α (X)的CoVaR不是相关参数的递增函数。类似的问题也出现在双变量t模型和有t个边距和一个Gumbel copula的模型中。在所有这些情况下，基于X≥VaR α (X)的CoVaR是依赖参数的递增函数。
{"title":"On dependence consistency of CoVaR and some other systemic risk measures","authors":"Georg Mainik, E. Schaanning","doi":"10.1515/strm-2013-1164","DOIUrl":"https://doi.org/10.1515/strm-2013-1164","url":null,"abstract":"Abstract This paper is dedicated to the consistency of systemic risk measures with respect to stochastic dependence. It compares two alternative notions of Conditional Value-at-Risk (CoVaR) available in the current literature. These notions are both based on the conditional distribution of a random variable Y given a stress event for a random variable X , but they use different types of stress events. We derive representations of these alternative CoVaR notions in terms of copulas, study their general dependence consistency and compare their performance in several stochastic models. Our central finding is that conditioning on X ≥ VaR α ( X ) gives a much better response to dependence between X and Y than conditioning on X = VaR α ( X ). We prove general results that relate the dependence consistency of CoVaR using conditioning on X ≥ VaR α ( X ) to well established results on concordance ordering of multivariate distributions or their copulas. These results also apply to some other systemic risk measures, such as the Marginal Expected Shortfall (MES) and the Systemic Impact Index (SII). We provide counterexamples showing that CoVaR based on the stress event X = VaR α ( X ) is not dependence consistent. In particular, if ( X , Y ) is bivariate normal, then CoVaR based on X = VaR α ( X ) is not an increasing function of the correlation parameter. Similar issues arise in the bivariate t model and in the model with t margins and a Gumbel copula. In all these cases, CoVaR based on X ≥ VaR α ( X ) is an increasing function of the dependence parameter.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2012-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/strm-2013-1164","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67313386","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In the conditional setting we provide a complete duality between quasiconvex risk measures defined on L 0 modules of the L p type and the appropriate class of dual functions. This is based on a general result which extends the usual Penot-Volle representation for quasiconvex real valued maps.
{"title":"Complete duality for quasiconvex dynamic risk measures on modules of the L p -type","authors":"M. Frittelli, Marco Maggis","doi":"10.1515/strm-2013-1163","DOIUrl":"https://doi.org/10.1515/strm-2013-1163","url":null,"abstract":"Abstract In the conditional setting we provide a complete duality between quasiconvex risk measures defined on L 0 modules of the L p type and the appropriate class of dual functions. This is based on a general result which extends the usual Penot-Volle representation for quasiconvex real valued maps.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2012-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/strm-2013-1163","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67313369","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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