Multi-portfolio time consistency for set-valued convex and coherent risk measures

Equivalent characterizations of multi-portfolio time consistency are deduced for closed convex and coherent set-valued risk measures on $L^{p}({\varOmega,\mathcal{F},\mathbb{P}; \mathbb{R}^{d}})$ with image space in the power set of $L^{p}({\varOmega,\mathcal{F}_{t},\mathbb{P}; \mathbb{R}^{d}})$. In the convex case, multi-portfolio time consistency is equivalent to a cocycle condition on the sum of minimal penalty functions. In the coherent case, multi-portfolio time consistency is equivalent to a generalized version of stability of the dual variables. As examples, the set-valued entropic risk measure with constant risk aversion coefficient is shown to satisfy the cocycle condition for its minimal penalty functions; the set of superhedging portfolios is shown to have in markets with proportional transaction costs the stability property and to satisfy in markets with convex transaction costs the composed cocycle condition; and a multi-portfolio time-consistent version of the set-valued average value at risk, the composed AV@R, is given, and its dual representation deduced.