Abstracts

A worst case distribution is defined to be a distribution minimising, among the risk factor distributions satisfying some plausibility constraint, the expectation of some fixed random payoff. The plausibility of a risk factor distribution is quantified by a convex integral function. This includes the special cases of relative entropy, Bregman distance, and $f$-divergence.
An ($\epsilon$-$\gamma$)-almost worst case distribution is a risk factor distribution which violates the plausibility constraint at most by the amount $\gamma$ and for which the expected payoff is better than the worst case by no more than $\epsilon$. From a practical point of view the localisation of almost worst case distributions determines the efficiency of a hedge against the worst case distribution. We prove that almost worst case distributions cluster in the Bregman neighbourhood of a positive function, which may be interpreted as a worst case localiser. In regular cases, the worst case localiser is at the same time the worst case density. But it may also be the case that the infimum of payoff expectations over the plausible distributions is not achieved, in which case the worst case localiser is not a worst case density, and perhaps not even a density.