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.

中文翻译：

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具有受限规范集的内部因素模型中的上限风险

对于（部分指定的）内部风险因素模型，与一般风险因素模型相比，我们建立了高度简化的超模排序结果。这使我们能够利用由风险成分和系统性风险因素组成的约束规范集提供的依赖信息，得出风险因素模型中联合投资组合的有意义且改进的风险界限。我们主要比较结果的证据不是标准的。它基于上颌骨的上颌骨的网格上足骨近似和传质理论。实际市场数据的应用程序显示了对标准方法的显着改进。