Maxitive monetary risk measures: worst-case risk assessment and sharp large deviations

In decision making under uncertainty and risk, worst-case risk assessments are often conducted using maxitive monetary risk measures. In this article, we study maxitive monetary risk measures on the space $L^0$ of all random variables identified modulo almost sure equality. We prove that a monetary risk measure is maxitive and continuous from below if and only if it is a penalized maximum loss. Furthermore, we characterize the maximum loss as the unique maxitive and law-invariant monetary risk measure. We apply the results to large deviation theory by providing a general criterion to establish a sharp large deviation estimate for sequences of probability measures. We use these findings to provide a formula for the asymptotics of the distortion-exponential insurance premium principle under risk pooling.