Vector-Valued Coherent Risk Measure Processes

Abstract

Introduced by Artzner et al. (1998) the axiomatic characterization of a static coherent risk measure was extended by Jouini et al. (2004) in a multi-dimensional setting to the concept of vector-valued risk measures. In this paper, we propose a dynamic version of the vector-valued risk measures in a continuous-time framework. Particular attention is devoted to the choice of a convenient risk space. We provide dual characterization results, we study different notions of time consistency and we give examples of vector-valued risk measure processes.

A Comparison of Techniques for Dynamic Multivariate Risk Measures

“…Remark 2.4. The dual representation given in [38] for the static coherent case (and in [8] for the dynamic case, see section 2.2 below) uses a single dual variable. This set of dual variables from [38] is equivalent to w T t (Q, w) : (Q, w) ∈ W t , and as discussed in the proof of theorem 2.3 in [25], the dual representation (2.1) and (2.3) can be given by this set alone.…”

“…The measurable selector approach was proposed in [8] and is an extension of [38] to the dynamic framework. Only coherent risk measures are considered in this approach as the technique used to deduce the dual representation relies on coherency.…”

“…[42]) in the coherent and polyhedral convex case, respectively via an approximation algorithm in the convex case, see [44,24]. [8] extended the results of [38] for coherent risk measures to the dynamic case. We will call this the measurable selector approach as it considers the value of a risk measures as a random set, and then provides a primal and dual representation for the measurable selectors in that set.…”

A supermartingale relation for multivariate risk measures

“…Set-valued risk measures have been studied in a single period framework in, e.g., [43,38,35,37,14]. Dynamic, multi-period set-valued risk measures have been studied in, e.g., [21,23,22,24,8]. We will mostly follow the setting and notation of [21,23] in this paper.…”

A recursive algorithm for multivariate risk measures and a set-valued Bellman’s principle

“…Multivariate risk measures, also called set-valued risk measures, were introduced in a static one-period setting by Meddeb, Jouini, Touzi [15] in 2004, and further studied in [13,10,11,4], to consider risk evaluations of random vectors motivated by market models with transaction costs. Dynamic set-valued risk measures were presented in [7,9,8,2]. The set-valued version of time-consistency, called multi-portfolio time consistency, was introduced in [7] and it was shown to be equivalent to a recursive form for the risk measure.…”