Optimal Bounds for Integrals with Respect to Copulas and Applications
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We consider the integration of two-dimensional, piecewise constant functions with respect to copulas. By drawing a connection to linear assignment problems, we can give optimal upper and lower bounds for such integrals and construct the copulas for which these bounds are attained. Furthermore, we show how our approach can be extended in order to approximate extremal values in very general situations. Finally, we apply our approximation technique to problems in financial mathematics and uniform distribution theory, such as the model-independent pricing of first-to-default swaps.
KeywordsLinear assignment problems Copulas Fréchet-Hoeffding bounds Credit risk Uniform distribution theory
The authors would like to thank Prof. Robert Tichy from TU Graz and Prof. Oto Strauch from the Slovak Academy of Science for helpful remarks and suggestions. The authors are also indebted to two anonymous referees who helped to improve the paper.
The second author is funded by the fellowship of the Doctoral School in Mathematics and Computer Science of University of Calabria and is partially supported by the Austrian Science Fund (FWF): W1230, Doctoral Program “Discrete Mathematics”.
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