Multidimensional integration by parts formulas are valid under the standard assumption that one of the functions is continuous and the other has bounded Hardy-Krause variation. Motivated by several recently developed results in the probabilistic context of price and risk bounds, we provide a version of an integration by parts formula for Lebesgue integrals on a possibly unbounded domain where a componentwise left-/right-continuous and measure inducing function is integrated with respect to the signed measure induced by a measure inducing, componentwise right-/left-continuous, bounded, and grounded function. For this purpose, we extensively analyze measure inducing functions and characterize them in terms of ∆-monotone functions. We provide, as a consequence of the integration by parts formula, several convergence results which allow an extension of the Lebesgue integral of a measure inducing function to the case that one integrates with respect to a Lipschitzcontinuous semi-copula.
Conditionally comonotonic risk vectors have been proved in [4] to yield worst case dependence structures maximizing the risk of the portfolio sum in partially specified risk factor models. In this paper we investigate the question how risk bounds depend on the specification of the pairwise copulas of the risk components Xiwith the systemic risk factor. As basic toolwe introduce a new ordering based on sign changes of the derivatives of copulas. This together with discretization by n-grids and the theory of supermodular transfers allows us to derive concrete ordering criteria for the maximal risks.
For the class of (partially specified) internal risk factor models we establish strongly simplified supermodular ordering results in comparison to the case of general risk factor models. This allows us to derive meaningful and improved risk bounds for the joint portfolio in risk factor models with dependence information given by constrained specification sets for the copulas of the risk components and the systemic risk factor. The proof of our main comparison result is not standard. It is based on grid copula approximation of upper products of copulas and on the theory of mass transfers. An application to real market data shows considerable improvement over the standard method.
We consider a completely specified factor model for a risk vector X = (X
1, . . ., Xd
), where the joint distributions of the components of X with a risk factor Z and the conditional distributions of X given Z are specified. We extend the notion of *-product of d-copulas as introduced for d = 2 and continuous factor distribution in Darsow et al. [6] and Durante et al. [8] to the multivariate and discontinuous case. We give a Sklar-type representation theorem for factor models showing that these *-products determine the copula of a completely specified factor model. We investigate in detail approximation, transformation, and ordering properties of *-products and, based on them, derive general orthant ordering results for completely specified factor models in dependence on their specifications. The paper generalizes previously known ordering results for the worst case partially specified risk factor models to some general classes of positive or negative dependent risk factor models. In particular, it develops some tools to derive sharp worst case dependence bounds in subclasses of completely specified factor models.
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