Time consistency for scalar multivariate risk measures

In this paper we present results on dynamic multivariate scalar risk measures, which
arise in markets with transaction costs and systemic risk. Dual representations of such risk measures are presented. These are then used to obtain the main results of this paper on time consistency; namely, an equivalent recursive formulation of multivariate scalar risk measures to multiportfolio time consistency. We are motivated to study time consistency of multivariate scalar risk measures as the superhedging risk measure in markets with transaction costs (with a single eligible asset) does not satisfy the usual scalar concept of time consistency. In fact,
scalar risk measures with the same scalarization weight at all times would not be
time consistent in general. The deduced recursive relation for the scalarizations of
multiportfolio time consistent set-valued risk measures provided in this paper requires consideration of the entire family of scalarizations. In this way we develop a direct notion of a “moving scalarization” for scalar time consistency that corroborates recent research on scalarizations of dynamic multi-objective problems.