Evaluations of Risk Measures for Different Probability Measures

Stochastic optimization problems, which arise in different areas, such as simple asset allocation problems or problems in insurance, often involve coherent risk measures. In these real-world problems the considered risk measure is frequently built on empirical distributions. It is therefore of interest to understand the potential deviation, which occurs when evaluating a risk measure for a perturbed, or slightly changed, probability measure. This paper addresses the potential deviation. It turns out that the Wasserstein distance, a well-known distance for probability measures, provides a valuable notion of distance in the present context: Many risk measures allow a precise quantification in terms of the Wasserstein distance, and important risk measures are continuous with respect to this distance. For specific random variables, which often occur in concrete, real-world problems, it is moreover demonstrated that the derived constants, describing the continuity properties, cannot be improved. The associated worst probability measures can be given explicitly in important situations.