  The relationship between set-valued risk measures for processes and vectors
on the optional filtration is investigated. The equivalence of risk measures
for processes and vectors and the equivalence of their penalty function
formulations are provided. In contrast with scalar risk measures, this
equivalence requires an augmentation of the set-valued risk measures for
processes. We utilize this result to deduce a new dual representation for risk
measures for processes in the set-valued framework. Finally, the equivalence of
multiportfolio time consistency between set-valued risk measures for processes
and vectors are provided; to accomplish this, an augmented definition for
multiportfolio time consistency of set-valued risk measures for processes is
proposed.
