Strongly Consistent Multivariate Conditional Risk Measures

TL;DR

This paper characterizes strongly consistent multivariate conditional risk measures, showing they decompose into aggregation functions and univariate risk measures, and are equivalent to multivariate certainty equivalents under law-invariance.

Contribution

It provides a decomposition theorem for multivariate conditional risk measures and links strong consistency with multivariate certainty equivalents.

Findings

Decomposition into aggregation and univariate risk measures

Strong consistency implies multivariate certainty equivalents

Law-invariance characterizes the risk measures as certainty equivalents

Abstract

We consider families of strongly consistent multivariate conditional risk measures. We show that under strong consistency these families admit a decomposition into a conditional aggregation function and a univariate conditional risk measure as introduced Hoffmann et al. (2016). Further, in analogy to the univariate case in F\"ollmer (2014), we prove that under law-invariance strong consistency implies that multivariate conditional risk measures are necessarily multivariate conditional certainty equivalents.