Multiportfolio time consistency for set-valued convex and coherent risk measures

TL;DR

This paper characterizes multiportfolio time consistency for set-valued convex and coherent risk measures, linking it to cocycle conditions and stability of dual variables, with applications to entropic risk, superhedging, and AV@R.

Contribution

It provides new equivalent characterizations of multiportfolio time consistency for set-valued risk measures, connecting it to cocycle conditions and dual variable stability.

Findings

Set-valued entropic risk measure satisfies cocycle condition.

Superhedging portfolios exhibit stability property.

Constructed a multiportfolio time consistent AV@R and derived its dual.

Abstract

Equivalent characterizations of multiportfolio time consistency are deduced for closed convex and coherent set-valued risk measures on $L^p(\Omega,\mathcal F, P; R^d)$ with image space in the power set of $L^p(\Omega,\mathcal F_t,P;R^d)$. In the convex case, multiportfolio time consistency is equivalent to a cocycle condition on the sum of minimal penalty functions. In the coherent case, multiportfolio 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 in markets with proportional transaction costs is shown to have the stability property and in markets with convex transaction costs is shown to satisfy the composed cocycle condition, and a multiportfolio time consistent version of the set-valued average value at risk, the composed AV@R, is given and its dual representation deduced.