When Can a Risk Measure Be Updated Consistently?

In this paper we aim at finding conditions under which risk measures can be consistently updated. Since the class of risk measures that allow dynamically consistent updates appears to be limited, we consider weaker notions of consistency in addition to dynamic consistency, concentrating on notions that are still strong enough to ensure uniqueness of updating. We show that a weak type of consistency called conditional consistency does have the uniqueness property, and we identify an update operator that must produce the consistent update if there exists such an update. We give necessary as well as sufficient conditions for the existence of consistent updates of a given risk measure, in the sense of conditional consistency and in related senses, and we clarify the relations between several notions of consistency. The theory is illustrated by examples of existence and nonexistence of consistent updates.