Vector-valued coherent risk measures

We define (d,n)-coherent risk measures as set-valued maps from <InlineEquation ID="Equ1"> <EquationSource Format="TEX">$L^\infty_d$</EquationSource> </InlineEquation> into <InlineEquation ID="Equ2"> <EquationSource Format="TEX">$\mathbb{R}^n$</EquationSource> </InlineEquation> 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 <InlineEquation ID="Equ3"> <EquationSource Format="TEX">$\mathbb{R}^d-$</EquationSource> </InlineEquation>valued random portfolio to <InlineEquation ID="Equ4"> <EquationSource Format="TEX">$\mathbb{R}^n-$</EquationSource> </InlineEquation>valued measure of risk. Necessary and sufficient conditions of coherent aggregation are provided. Copyright Springer-Verlag Berlin/Heidelberg 2004