Time consistency for scalar multivariate risk measures

TL;DR

This paper develops a recursive formulation for dynamic multivariate scalar risk measures, establishing a notion of 'moving scalarization' that ensures time consistency in markets with transaction costs and systemic risk.

Contribution

It introduces a new recursive relation for scalarizations of multivariate risk measures, enabling time consistency through a dynamic 'moving scalarization' approach.

Findings

Provides dual representations of multivariate scalar risk measures.

Establishes an equivalent recursive formulation for multiportfolio time consistency.

Introduces the concept of 'moving scalarization' for scalar time consistency.

Abstract

In this paper we present results on dynamic multivariate scalar risk measures, which arise in markets with transaction costs and systemic risk. Dual representations of such risk measures are presented. These are then used to obtain the main results of this paper on time consistency; namely, an equivalent recursive formulation of multivariate scalar risk measures to multiportfolio time consistency. We are motivated to study time consistency of multivariate scalar risk measures as the superhedging risk measure in markets with transaction costs (with a single eligible asset) (Jouini and Kallal (1995), Roux and Zastawniak (2016), Loehne and Rudloff (2014)) does not satisfy the usual scalar concept of time consistency. In fact, as demonstrated in (Feinstein and Rudloff (2021)), scalar risk measures with the same scalarization weight at all times would not be time consistent in general. The deduced recursive relation for the scalarizations of multiportfolio time consistent set-valued risk measures provided in this paper requires consideration of the entire family of scalarizations. In this way we develop a direct notion of a "moving scalarization" for scalar time consistency that corroborates recent research on scalarizations of dynamic multi-objective problems (Karnam, Ma, and Zhang (2017), Kovacova and Rudloff (2021)).