Calibrating reference measures of law-invariant functionals
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. 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. While this stance is in contrast to wide parts of the literature that treat the reference probability as given, it is instead more in line with the investigation of probabilistically sophisticated preferences. Concerning question (i), we 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. Concerning the calibration, we show how to infer the reference measure as a related supremum or infimum in the space of bounded charges. While it is generally versatile, this approach fails in the important case of the Value-at-Risk. Here, a suitable alternative is presented.
Area: CS14 - Stochastic models for risk and cooperation (Giacomo Scandolo and Alessandro Doldi)
Keywords: Law invariance, risk measures, reference probabilities, Value-at-Risk
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