Optimal Bounds for Integrals with Respect to Copulas and Applications

Abstract

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.

Distribution functions, extremal limits and optimal transport

“…An approximation result suited for the uniform marginals situation is Theorem 2.2 from [25], here we can complement it by proving that a (sub-)sequence of the discrete optimizers converges to an optimizer of the limiting continuous problem.…”

“…Proof: The first statement (13) is already shown in [25]. For the remaining part we can proceed in the spirit of Villani's proof of Theorem 5.20 from [49], notice there a sequence of continuous functions c n is considered.…”

“…Since the computational methods are based on the assignment problem, see Burkard et al [13], we recapitulate it and mention (some) one of the fundamental numerical solution algorithm(s). Some connections between the particular copula maximization problem and the assignment problem are already given in [25]. There the authors showed that for a piecewise constant cost function the maximizing measure is induced by a shuffle of M whose parameters are linked to the permutation which solves the corresponding assignment problem.…”

“…There the marginal distributions are fixed to be uniform, whereas the cost function is approximated by piecewise constant functions on a deterministically chosen grid. This particular situation is linked to a solution of the assignment problem via Theorem 2.1 of [25].…”

“…This last problem is not easy to handle with and, apart from the already mentioned papers [23,35], only [22] is known, where the authors found an explicit asymptotic distribution function of the sequence (φ b (n), φ b (n + 1), φ b (n + 2)) n≥0 . Problem (2) has been recently studied in [25] in connection with a well-known problem in combinatorial optimization, namely the linear assignment problem. By means of this tool, the authors give optimal upper and lower bounds for integrals of two-dimensional, piecewise constant functions with respect to copulas and construct the copulas for which these bounds are attained.…”