Weakly time consistent concave valuations and their dual representations

We derive dual characterizations of two notions of weak time consistency for concave valuations, which are convex risk measures under a positive sign convention. Combined with a suitable risk aversion property, these notions are shown to amount to three simple rules for not necessarily minimal representations, describing precisely which features of a valuation determine its unique consistent update. A compatibility result shows that for a time-indexed sequence of valuations, it is sufficient to verify these rules only pairwise with respect to the initial valuation, or in discrete time, only stepwise. We conclude by describing classes of consistently risk averse dynamic valuations with prescribed static properties per time step. This gives rise to a new formalism for recursive valuation.