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In decision making under uncertainty and risk, worst-case risk assessmentsare often conducted using maxitive monetary risk measures. In this article, westudy maxitive monetary risk measures on the space $L^0$ of all randomvariables identified modulo almost sure equality. We prove that a monetary riskmeasure is maxitive and continuous from below if and only if it is a penalizedmaximum loss. Furthermore, we characterize the maximum loss as the uniquemaxitive and law-invariant monetary risk measure. We apply the results to largedeviation theory by providing a general criterion to establish a sharp largedeviation estimate for sequences of probability measures. We use these findingsto provide a formula for the asymptotics of the distortion-exponentialinsurance premium principle under risk pooling.

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