On set-valued functionals: multivariate risk measures and Aumann integrals

Abstract

In this dissertation, multivariate risk measures for random vectors and Aumann integrals of set-valued functions are studied. Both are set-valued functionals with values in a complete lattice of subsets of $\mathbb{R}^m$.

Multivariate risk measures are considered in a general $d$-asset financial market with trading opportunities in discrete time. Specifically, the following features of the market are incorporated in the evaluation of multivariate risk: convex transaction costs modeled by solvency regions, intermediate trading constraints modeled by convex random sets, and the requirement of liquidation into the first $m \leq d$ of the assets. It is assumed that the investor has a ``pure" multivariate risk measure $R$ on the space of $m$-dimensional random vectors which represents her risk attitude towards the assets but does not take into account the frictions of the market. Then, the investor with a $d$-dimensional position minimizes the set-valued functional $R$ over all $m$-dimensional positions that she can reach by trading in the market subject to the frictions described above. The resulting functional $R^{\text{mar}}$ on the space of $d$-dimensional random vectors is another multivariate risk measure, called the market-extension of $R$. A dual representation for $R^{\text{mar}}$ that decomposes the effects of $R$ and the frictions of the market is proved.

Next, multivariate risk measures are studied in a utility-based framework. It is assumed that the investor has a complete risk preference towards each individual asset, which can be represented by a von Neumann-Morgenstern utility function. Then, an incomplete preference is considered for multivariate positions which is represented by the vector of the individual utility functions. Under this structure, multivariate shortfall and divergence risk measures are defined as the optimal values of set minimization problems. The dual relationship between the two classes of multivariate risk measures is constructed via a recent Lagrange duality for set optimization. In particular, it is shown that a shortfall risk measure can be written as an intersection over a family of divergence risk measures indexed by a scalarization parameter. Examples include the multivariate versions of the entropic risk measure and the average value at risk.

In the second part, Aumann integrals of set-valued functions on a measurable space are viewed as set-valued functionals and a Daniell-Stone type characterization theorem is proved for such functionals. More precisely, it is shown that a functional that maps measurable set-valued functions into a certain complete lattice of subsets of $\mathbb{R}^m$ can be written as the Aumann integral with respect to a measure if and only if the functional is (1) additive and (2) positively homogeneous, (3) it preserves decreasing limits, (4) it maps halfspace-valued functions to halfspaces, and (5) it maps shifted cone-valued functions to shifted cones. While the first three properties already exist in the classical Daniell-Stone theorem for the Lebesgue integral, the last two properties are peculiar to the set-valued framework and they suffice to complement the first three properties to identify a set-valued functional as the Aumann integral with respect to a measure.