Sharp Bounds on a Class of Copulas with known Values at Several Points

Sadooghi-Alvandi, S. M.; Shishebor, Z.; Mardani-Fard, Heydar Ali

doi:10.1080/03610926.2011.607529

Cited by 8 publications

(7 citation statements)

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“…[25] proves that if is increasing, then B Q is a copula and if is decreasing, then A Q is a copula. Thus, in Theorem 2.1 of Sadooghi-Alvandi et al [23], is increasing, C is obviously not empty. Similarly, when is decreasing which is indicated in Theorem 2.4 of Sadooghi-Alvandi et al [23], C is also not empty.…”

Section: The Infimum Of Functions With the 1-lipschitz Property Also mentioning

confidence: 97%

“…Thus, in Theorem 2.1 of Sadooghi-Alvandi et al [23], is increasing, C is obviously not empty. Similarly, when is decreasing which is indicated in Theorem 2.4 of Sadooghi-Alvandi et al [23], C is also not empty. Theorem 2.2 and 2.3 of Sadooghi-Alvandi et al [23] assume that is increasing or decreasing thus the same conclusion as in Tankov [25] is obtained.…”

Section: The Infimum Of Functions With the 1-lipschitz Property Also mentioning

confidence: 97%

“…This section is an extension of Theorem 1 of Tankov [25] and Sadooghi-Alvandi et al [23]. To prove our results, we use the following well-known lemma.…”

Section: Sufficient Conditions For Both Lower and Upper Bounds To Be mentioning

confidence: 99%

“…The purpose of this section is to illustrate how to construct the best bounds in the sense of the infinite norm in the case when there are two constraints. Sadooghi-Alvandi et al [23] also solve this problem but assume that the two constraints are either non-decreasing or non-increasing and thus are special cases of Tankov [25] main result. 5 Hereafter, we derive explicit expressions for the smallest copula above the lower bound and the largest copula below the upper bound for the infinite norm.…”

Section: Compact Set With 2 Pointsmentioning

confidence: 99%

“…To do so, we make use of the recent work of Tankov [25] who generalize Nelsen [18]'s result by giving explicit expressions for the best upper and lower bounds for a bivariate copula when its values on a compact subset of [0 1] 2 are known (see also Sadooghi-Alvandi et al [23]). Tankov shows that they are quasi-copulas and not necessarily copulas.…”

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Bounds on Capital Requirements For Bivariate Risk with Given Marginals and Partial Information on the Dependence

Bernard¹,

Liu

MacGillivray

et al. 2013

Dependence Modeling

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Nelsen et al. [20] find bounds for bivariate distribution functions when there are constraints on the values of its quartiles. Tankov [25] generalizes this work by giving explicit expressions for the best upper and lower bounds for a bivariate copula when its values on a compact subset of [0 1] 2 are known. He shows that they are quasi-copulas and not necessarily copulas. Tankov [25] and Bernard et al. [3] both give sufficient conditions for these bounds to be copulas. In this note we give weaker sufficient conditions to ensure that both bounds are simultaneously copulas. Furthermore, we develop a novel application to quantitative risk management by computing bounds on a bivariate risk measure. This can be useful in optimal portfolio selection, in reinsurance, in pricing bivariate derivatives or in determining capital requirements when only partial information on dependence is available. Nelsen [18] derives best possible bounds when the copula is known at a specific point. Our objective in this paper is to extend this literature to the case when the copula is known in more than one point and to show how these bounds can be useful in quantifying dependence misspecification. Assuming that marginals are given and that the dependence is unspecified, or partly unspecified, bounds on copulas can be indeed used to quantify this type of model risk. ). Tankov shows that they are quasi-copulas and not necessarily copulas. In this paper, we focus on deriving bounds on copulas that are also copulas. The first section focuses on finding simple conditions to ensure that Tankov's bounds are copulas. When the bounds are not copulas, it is possible to approximate them by a copula. The second section illustrates a method to find the best copula for the uniform norm that approximates the * Keywords

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Section: The Infimum Of Functions With the 1-lipschitz Property Also mentioning

confidence: 97%

Section: The Infimum Of Functions With the 1-lipschitz Property Also mentioning

confidence: 97%

“…This section is an extension of Theorem 1 of Tankov [25] and Sadooghi-Alvandi et al [23]. To prove our results, we use the following well-known lemma.…”

Section: Sufficient Conditions For Both Lower and Upper Bounds To Be mentioning

confidence: 99%

Section: Compact Set With 2 Pointsmentioning

confidence: 99%

mentioning

confidence: 99%

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