Partial Law Invariance and Risk Measures

We introduce the concept of partial law invariance, generalizing the concepts of law invariance and probabilistic sophistication widely used in decision theory, as well as 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. Strong partial law invariance is defined to bridge the gap between the above characterization and the classic representation formula of Kusuoka. 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 different types of uncertainty. We provide a tractable optimization formula for computing a class of partially law-invariant coherent risk measures and give a numerical example.