fréchet bounds
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2019 ◽  
Vol 10 (1) ◽  
pp. 1-12
Author(s):  
Caroline Uhler ◽  
Donald Richards

We consider the lattice, $\mathcal{L}$, of all subsets of a multidimensional contingency table and establish the properties of monotonicity and supermodularity for the marginalization function, $n(\cdot)$, on $\mathcal{L}$.  We derive from the supermodularity of $n(\cdot)$ some generalized Fr\'echet inequalities complementing and extending inequalities of Dobra and Fienberg.  Further, we construct new monotonic and supermodular functions from $n(\cdot)$, and we remark on the connection between supermodularity and some correlation inequalities for probability distributions on lattices.  We also apply an inequality of Ky Fan to derive a new approach to Fr\'echet inequalities for multidimensional contingency tables.


2016 ◽  
Vol 4 (1) ◽  
Author(s):  
Giovanni Puccetti ◽  
Ludger Rüschendorf ◽  
Dennis Manko

AbstractBased on a novel extension of classical Hoeffding-Fréchet bounds, we provide an upper VaR bound for joint risk portfolios with fixed marginal distributions and positive dependence information. The positive dependence information can be assumed to hold in the tails, in some central part, or on a general subset of the domain of the distribution function of a risk portfolio. The newly provided VaR bound can be interpreted as a comonotonic VaR computed at a distorted confidence level and its quality is illustrated in a series of examples of practical interest.


2012 ◽  
Vol 49 (3) ◽  
pp. 866-875 ◽  
Author(s):  
Carole Bernard ◽  
Xiao Jiang ◽  
Steven Vanduffel

Tankov (2011) improved the Fréchet bounds for a bivariate copula when its values on a compact subset of [0, 1]2 are given. He showed that the best possible bounds are quasi-copulas and gave a sufficient condition for these bounds to be copulas. In this note we give weaker sufficient conditions to ensure that the bounds are copulas. We also show how this can be useful in portfolio selection. It turns out that finding a copula as a lower bound plays a key role in determining optimal investment strategies explicitly for investors with some type of state-dependent constraints.


2012 ◽  
Vol 49 (03) ◽  
pp. 866-875 ◽  
Author(s):  
Carole Bernard ◽  
Xiao Jiang ◽  
Steven Vanduffel

Tankov (2011) improved the Fréchet bounds for a bivariate copula when its values on a compact subset of [0, 1]2 are given. He showed that the best possible bounds are quasi-copulas and gave a sufficient condition for these bounds to be copulas. In this note we give weaker sufficient conditions to ensure that the bounds are copulas. We also show how this can be useful in portfolio selection. It turns out that finding a copula as a lower bound plays a key role in determining optimal investment strategies explicitly for investors with some type of state-dependent constraints.


Author(s):  
THOMAS FETZ

This article is devoted to the propagation of families of variability intervals through multivariate functions comprising the semantics of confidence limits. At fixed confidence level, local random sets are defined whose aggregation admits the calculation of upper probabilities of events. In the multivariate case, a number of ways of combination is highlighted to encompass independence and unknown interaction using random set independence and Fréchet bounds. For all cases we derive formulas for the corresponding upper probabilities and elaborate how they relate. An example from structural mechanics is used to exemplify the method.


2011 ◽  
Vol 48 (02) ◽  
pp. 389-403 ◽  
Author(s):  
Peter Tankov

Improved bounds on the copula of a bivariate random vector are computed when partial information is available, such as the values of the copula on a given subset of [0, 1]2, or the value of a functional of the copula, monotone with respect to the concordance order. These results are then used to compute model-free bounds on the prices of two-asset options which make use of extra information about the dependence structure, such as the price of another two-asset option.


2011 ◽  
Vol 48 (2) ◽  
pp. 389-403 ◽  
Author(s):  
Peter Tankov

Improved bounds on the copula of a bivariate random vector are computed when partial information is available, such as the values of the copula on a given subset of [0, 1]2, or the value of a functional of the copula, monotone with respect to the concordance order. These results are then used to compute model-free bounds on the prices of two-asset options which make use of extra information about the dependence structure, such as the price of another two-asset option.


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