Title: Time consistency for scalar multivariate risk measures,
Abstract: 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) (Jouini and Kallal
(1995), Roux and Zastawniak (2016), Loehne and Rudloff (2014)) does not satisfy
the usual scalar concept of time consistency. In fact, as demonstrated in
(Feinstein and Rudloff (2018)), 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 (Karnam, Ma, and Zhang (2017), Kovacova and Rudloff (2018)).