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The relationship between set-valued risk measures for processes and vectorson the optional filtration is investigated. The equivalence of risk measuresfor processes and vectors and the equivalence of their penalty functionformulations are provided. In contrast with scalar risk measures, thisequivalence requires an augmentation of the set-valued risk measures forprocesses. We utilize this result to deduce a new dual representation for riskmeasures for processes in the set-valued framework. Finally, the equivalence ofmultiportfolio time consistency between set-valued risk measures for processesand vectors is provided; to accomplish this, an augmented definition formultiportfolio time consistency of set-valued risk measures for processes isproposed.

Set-Valued Dynamic Risk Measures for Processes and Vectors