Multivariate risk measures: a constructive approach based on selections (Q2831005)

The authors explore a framework in which set-valued risk measures on set-valued portfolios allow for the description and analysis of the risk of portfolios for which profits/losses of the portfolio constituents -- e.g.\ business lines of a company, or subportfolios corresponding to currencies or geographic locations -- may not offset each other e.g.\ due to transaction costs, taxes or other fees, or because of regulatory reasons or other restrictions.NEWLINENEWLINEA \textit{set-valued portfolio} is an almost surely nonempty random closed convex subset of \(\mathbb{R}^d\). A \textit{selection} of a set-valued portfolio \(X\) is a random vector \(\xi\) which satisfies \(\xi \in X\) almost surely. A function \(\rho\) mapping set-valued portfolios to upper convex sets is a \textit{set-valued coherent risk measure} if {\parindent=0.7cmNEWLINE\begin{itemize}\item[--] \(\rho(X+a) = \rho(X) - a\)\quad (\(a \in \mathbb{R}\));NEWLINE\item[--] if \(X \subseteq Y\) almost surely then \(\rho(X) \subseteq \rho(Y)\);NEWLINE\item[--] \(\rho(c \cdot X) = c\cdot \rho(X)\)\quad \((c >0)\);NEWLINE\item[--] \(\rho(X+Y) \supseteq \rho(X) + \rho(Y)\). NEWLINENEWLINE\end{itemize}}NEWLINEA set-valued portfolio \(X\) is called \textit{acceptable} if \(0 \in \rho(X)\). For a random vector \(Z = (\zeta_{i})_{i=1, \dots, d} \in \mathbb{R}^d\), \(Z \cdot X\) is the set-valued portfolio obtained from \(X\) by scaling its \(i\)th coordinate by \(\zeta_{i}\) (\(i=1, \dots, d\)).NEWLINENEWLINELet \(\{r_{i} : i=1, \dots, d\}\) be (real-valued) coherent risk measures. For a random vector \(\xi = (\xi_{1},\dots,\xi_{d}) \in \mathbb{R}^d\), set \(r(\xi) = (r_{1}(\xi_{1}),\dots,r_{d}(\xi_{d}))\). Then \(\xi\) is \textit{acceptable} if \(r(\xi) \leq 0\); and a set-valued portfolio \(X\) is \textit{selection acceptable} if it has an acceptable selection. The authors study in detail \textit{selection risk measures}, defined as NEWLINE\[NEWLINE\rho(X) = \{x \in \mathbb{R}^d : X+x \text{ is selection acceptable}\}.NEWLINE\]NEWLINE The approximation and computation of selection risk measures is also discussed in the special case when \(X\) follows a bivariate normal distribution.NEWLINENEWLINEFor every nonempty set \(\mathcal{Z}\) of nonnegative \(q\)-integrable random vectors \(Z \in \mathbb{R}^d\), a set-valued coherent risk measures can be defined as NEWLINE\[NEWLINE\rho_{\mathcal{Z}}(X) = \bigcap_{Z\in \mathcal{Z}} \frac{\mathbb{E}[\hat X Z]}{\mathbb{E}[Z]},NEWLINE\]NEWLINE where \(\mathbb{E}[\hat X Z]\) is the closure of the set \(\{\mathbb{E}[\xi] : \xi \text{ is an integrable selection of } Z \cdot X\}\). The authors also obtain that every set-valued coherent risk measure satisfying some continuity property can be obtained in this form.