Distributional robustness, stochastic divergences, and the quadrangle of risk

In the distributional robustness approach to optimization under uncertainty, ambiguity about which probability distribution to use is addressed by turning to the worst that might occur with respect to a specified set of alternative probability distributions. Such sets are often taken to be neighborhoods of some nominal distribution with respect to a stochastic divergence like that of Kullback–Leibler or Wasserstein. Here that approach is coordinated with the fundamental quadrangle of risk with its quantifications not only of risk, but also regret, deviation and error, along with the functionals that dualize them. Stochastic divergences are introduced axiomatically and shown to constitute the duals of risk measures in a special class. Rules are uncovered for how regret measures for those risk measures can be obtained by appropriate extensions of the divergence functional. This reveals clearly the pattern in which the robustness functionals coming from divergence neighborhoods can be provided with other formulas featuring minimization instead of maximization, which is beneficial for optimization schemes. To get everything to fit, however the aversity properties of risk and the rest that, until now, have been imposed in the quadrangle of relationships must be relaxed. A suitable substitute, called subaversity, is found that works while only differing from aversity for functionals that are not positively homogeneous.