The (?) reference measure

A functional on random variables is law invariant with respect to a reference probability (or probabilistically sophisticated) if its value only depends on the distribution of its argument under that measure. The class of such functionals is not only vast, but of integral importance for financial, actuarial, and economic applications. In this talk, we take a concrete functional as given and ask (i) if there can be more than one such reference probability, and (ii) how one can infer the reference probability from the functional. This stance is in contrast to wide parts of the literature that treat the reference probability as given. Nevertheless, as I will show at the beginning of my talk, it is in line with the investigation of probabilistically sophisticated preferences and arises as a natural consequence of the Marinacci-Svistula Uniqueness Theorem. Concerning question (i), I demonstrate that uniqueness holds for a wide class of functionals unless they are constant or depend only on the essential supremum and essential infimum of the argument. Applications to monetary and return risk measures will be given. Concerning the calibration question (ii), I show how to infer the reference measure as a related supremum or infimum in the space of bounded charges. While this approach is generally versatile, it fails in the important case of the Value-at-Risk. Here, a suitable alternative is presented.