Membership conditions for consistent families of monetary valuations

We investigate time consistency of monetary valuations, also called monetary risk measures or monetary utility functions. Through a number of recent research contributions, it has become clear that time consistency imposes strong constraints on families of monetary valuations conditioned on available information at different time instants. In this paper we add to these results by showing that consistent families of monetary valuations are already determined uniquely by the choice of the initial valuation, under suitable sensitivity assumptions; moreover, this statement holds even when the term “consistency” is interpreted in a rather weak sense. The unique update rule is specified explicitly, and we characterize the existence of consistent updates for a given initial monetary valuation. We give examples of situations in which weak consistency is relevant. An application is given to the construction of consistent families of compound valuations, as an illustration of the additional flexibility under weak time consistency