Time consistency for scalar multivariate risk measures

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), Löhne and Rudloff (2014), Roux and Zastawniak (2016)) 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), Kováčová and Rudloff (2021)).

Complex Risk Statistics with Scenario Analysis

Complex risk is a critical factor for both intelligent systems and risk management. In this paper, we consider a special class of risk statistics, named complex risk statistics. Our result provides a new approach for addressing complex risk, especially in deep neural networks. By further developing the properties related to complex risk statistics, we are able to derive dual representations for such risk.

EXPRESS: Top-down influences on working memory representations of faces: Evidence from dual-target visual search

Variability in appearance across different images of the same unfamiliar face often causes participants to perceive different faces. Because perceptual information is not sufficient to link these encounters, top-down guidance may be critical in the initial stages of face learning. Here we examine the interaction between top-down guidance and perceptual information when forming memory representations of unfamiliar faces. In two experiments, we manipulated the names associated with images of a target face that participants had to find in a search array. In Experiment 1, wrongly labelling two images of the same face with different names resulted in more errors relative to when the faces were labelled correctly. In Experiment 2, we compared this cost of mislabelling to the established ‘dual-target search cost’ where searching for two targets produces more search errors relative to one target. We found search costs when searching for two different faces, but not when searching for mislabelled images of the same face. Together, these results suggest that perceptual and semantic information interact when we in the form face memory representations Mislabelling the identity of perceptually similar faces does not cause dual representations to be created, but rather that it impedes the process of forming a single robust representation.

Regulator-Based Risk Statistics with Scenario Analysis

When there are potential risks in the progress of the engineering project, regulators pay more attentions to losses rather than gains. In this paper, we design a new class of risk statistics for engineering, named regulator-based risk statistics. Considering the properties of regulator-based risk statistics, we are able to derive the dual representations for them. At last, the regulator-based version is investigated.

A Theory for Measures of Tail Risk

The notion of “tail risk” has been a crucial consideration in modern risk management and financial regulation, as very well documented in the recent regulatory documents. To achieve a comprehensive understanding of the tail risk, we carry out an axiomatic study for risk measures that quantify the tail risk, that is, the behaviour of a risk beyond a certain quantile. Such risk measures are referred to as tail risk measures in this paper. The two popular classes of regulatory risk measures in banking and insurance, value at risk (VaR) and expected shortfall, are prominent, yet elementary, examples of tail risk measures. We establish a connection between a tail risk measure and a corresponding law-invariant risk measure, called its generator, and investigate their joint properties. A tail risk measure inherits many properties from its generator, but not subadditivity or convexity; nevertheless, a tail risk measure is coherent if and only if its generator is coherent. We explore further relevant issues on tail risk measures, such as bounds, distortion risk measures, risk aggregation, elicitability, and dual representations. In particular, there is no elicitable tail convex risk measure other than the essential supremum, and under a continuity condition, the only elicitable and positively homogeneous monetary tail risk measures are the VaRs.

Nonlinear expectations of random sets

Sublinear functionals of random variables are known as sublinear expectations; they are convex homogeneous functionals on infinite-dimensional linear spaces. We extend this concept for set-valued functionals defined on measurable set-valued functions (which form a nonlinear space) or, equivalently, on random closed sets. This calls for a separate study of sublinear and superlinear expectations, since a change of sign does not alter the direction of the inclusion in the set-valued setting.We identify the extremal expectations as those arising from the primal and dual representations of nonlinear expectations. Several general construction methods for nonlinear expectations are presented and the corresponding duality representation results are obtained. On the application side, sublinear expectations are naturally related to depth trimming of multivariate samples, while superlinear ones can be used to assess utilities of multiasset portfolios.

Multi-utility representations of incomplete preferences induced by set-valued risk measures

We establish a variety of numerical representations of preference relations induced by set-valued risk measures. Because of the general incompleteness of such preferences, we have to deal with multi-utility representations. We look for representations that are both parsimonious (the family of representing functionals is indexed by a tractable set of parameters) and well behaved (the representing functionals satisfy nice regularity properties with respect to the structure of the underlying space of alternatives). The key to our results is a general dual representation of set-valued risk measures that unifies the existing dual representations in the literature and highlights their link with duality results for scalar risk measures.

Squiggly Economics

Three generic economic optimization problems (expenditure (cost) minimization, revenue maximization, and profit maximization) are studied using the mathematical tools developed in Chapters 2 and 3. Conjugate duality results are developed for each. The resulting dual representations (E(q;y), R(p,x), and π‎(p,q)) are shown to characterize all of the economically relevant information in, respectively, V(y), Y(x), and Gr(≽(y)). The implications of different restrictions on ≽(y) for the dual representations are examined.

Surplus-Invariant Risk Measures

This paper presents a systematic study of the notion of surplus invariance, which plays a natural and important role in the theory of risk measures and capital requirements. So far, this notion has been investigated in the setting of some special spaces of random variables. In this paper, we develop a theory of surplus invariance in its natural framework, namely, that of vector lattices. Besides providing a unifying perspective on the existing literature, we establish a variety of new results including dual representations and extensions of surplus-invariant risk measures and structural results for surplus-invariant acceptance sets. We illustrate the power of the lattice approach by specifying our results to model spaces with a dominating probability, including Orlicz spaces, as well as to robust model spaces without a dominating probability, where the standard topological techniques and exhaustion arguments cannot be applied.