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Bounds on capital requirements for bivariate risk with given marginals and partial information on the dependence. (English) Zbl 06297671

Summary: R. B. Nelsen et al. [J. Multivariate Anal. 90, No. 2, 348–358 (2004; Zbl 1057.62038)] find bounds for bivariate distribution functions when there are constraints on the values of its quartiles. P. Tankov [J. Appl. Probab. 48, No. 2, 389–403 (2011; Zbl 1219.60016)] 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 [loc. cit.] and C. Bernard et al. [J. Appl. Probab. 49, No. 3, 866–875 (2012; Zbl 1259.60022)] 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.

MSC:

62E99 Statistical distribution theory
62H99 Multivariate analysis
62P05 Applications of statistics to actuarial sciences and financial mathematics
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