Vector-valued coherent risk measures

Abstract.

We define (d,n)-coherent risk measures as set-valued maps from $L^\infty_d$ into $\mathbb{R}^n$ satisfying some axioms. We show that this definition is a convenient extension of the real-valued risk measures introduced by Artzner et al. [2]. We then discuss the aggregation issue, i.e., the passage from $\mathbb{R}^d-$valued random portfolio to $\mathbb{R}^n-$valued measure of risk. Necessary and sufficient conditions of coherent aggregation are provided.