Set-valued dynamic risk measures for processes and for vectors

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 to scalar risk measures, this equivalence requires an augmentation of the set-valued risk measures for processes. We utilise this result to deduce a new dual representation for risk measures for processes in the set-valued framework. Finally, the equivalence of multi-portfolio time-consistency between set-valued risk measures for processes and vectors is provided. To accomplish this, an augmented definition for multi-portfolio time-consistency of set-valued risk measures for processes is proposed.