Vector-valued coherent risk measures
The authors consider the situation where the risky portfolio is an \(\mathbb R^d\)-valued vector variable. It is assumed that a partial ordering on \(\mathbb R^d\) is given. A specification of this order accounts for some frictions on the financial market such as transaction costs, liquidity problems, irreversible transfers. Given an integer \(n\leq d\), the \((d,n)\)-coherent risk measure consistent with the order is defined as a set-valued map \(R\) from \(L_d^\infty\) into \(\mathbb R^n\) satisfying some convenient axioms. The geometric and topological properties implied by the definition of a vector-valued coherent risk measure are studied. Dual representations of the vector-valued coherent risk measures are provided. The definition of vector-valued coherent risk measures works when \(n<d\), i.e., the risk of a \(\mathbb R^d\)-valued random variable is required to be cancelled by \(\mathbb R^n\) deterministic portfolios. This is related to an aggregation problem. Two alternative methods of aggregation are studied: portfolio and risk aggregation. Necessary and sufficient conditions for the corresponding set-valued map to be a \((d,n)\)-coherent risk measure are established.
scientific article; zbMATH DE number 2169548