Partially Law-Invariant Risk Measures

We introduce the concept of partial law invariance, generalizing the concept of law-invariant risk measures widely used in statistical and financial applications. This new concept is motivated by practical considerations of decision-making under uncertainty, thus connecting the literature on decision theory and that on financial risk management. We fully characterize partially law-invariant coherent risk measures via a novel representation formula, which, surprisingly, has little resemblance to the classical formula for law-invariant coherent risk measures. A notion of strong partial law invariance is introduced, allowing for a representation formula akin to the classical one. We propose a few classes of new risk measures, including partially law-invariant versions of the Expected Shortfall and the entropic risk measures, and illustrate their applications in risk assessment under model uncertainty.