scholarly journals Set-Valued T-Translative Functions and Their Applications in Finance

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2270
Author(s):  
Andreas H. Hamel ◽  
Frank Heyde
Keyword(s):  
Hilbert Spaces ◽  
Risk Measures ◽  
Random Sets ◽  
Scalar Case ◽  
Infimal Convolution ◽  
Dual Representation ◽  
New Concepts ◽  
Wide Range ◽  
Primal Dual ◽  
Vector Valued

A theory for set-valued functions is developed, which are translative with respect to a linear operator. It is shown that such functions cover a wide range of applications, from projections in Hilbert spaces, set-valued quantiles for vector-valued random variables, to scalar or set-valued risk measures in finance with defaultable or nondefaultable securities. Primal, dual, and scalar representation results are given, among them an infimal convolution representation, which is not so well known even in the scalar case. Along the way, new concepts of set-valued lower/upper expectations are introduced and dual representation results are formulated using such expectations. An extension to random sets is discussed at the end. The principal methodology consisted of applying the complete lattice framework of set optimization.

scholarly journals TOWARDS FUNCTIONAL ARCHITECTURAL DECOMPOSITION OF CONSTRAINED LAYER INFLATABLE SYSTEMS

2021 ◽  
Vol 1 ◽  
pp. 3219-3228
Author(s):  
Koray Benli ◽  
Jonathan Luntz ◽  
Diann Brei ◽  
Wonhee Kim ◽  
Paul Alexander ◽  
...  
Keyword(s):  
Conceptual Design ◽  
Form And Function ◽  
New Concepts ◽  
Wide Range ◽  
Systematic Understanding ◽  
And Function

AbstractPneumatically activated systems enable myriad types of highly functional inflatables employing a wide range of architectural approaches affecting their form and function, making systematic conceptual design difficult. A new architectural class of pneumatically activated systems, constrained layer inflatable systems, consists of hierarchically architected flat layers of thin airtight bladders that are internally and/or externally constrained to generate a variety of functionalities. The highly hierarchical architectural structure of constrained layer inflatable systems coincides with the hierarchy of produced functions, providing an opportunity for the development of a functional architectural decomposition, capturing the inherent relationship between architectural and functional hierarchies. The basis of the approach is conveyed through the design of an example constrained layer inflatable system. This approach empowers the systematic understanding of the interrelated architectural and functional breakdown of constrained layer inflatable systems, enabling designers to iteratively analyze, synthesize, and re-synthesize the components of the system improving existing designs and exploring new concepts.

scholarly journals Multivariate extensions of expectiles risk measures

2017 ◽  
Vol 5 (1) ◽  
pp. 20-44 ◽  
Author(s):  
Véronique Maume-Deschamps ◽  
Didier Rullière ◽  
Khalil Said
Keyword(s):  
Stochastic Approximation ◽  
Risk Measures ◽  
New Family ◽  
Vector Valued

Abstract This paper is devoted to the introduction and study of a new family of multivariate elicitable risk measures. We call the obtained vector-valued measures multivariate expectiles. We present the different approaches used to construct our measures. We discuss the coherence properties of these multivariate expectiles. Furthermore, we propose a stochastic approximation tool of these risk measures.

scholarly journals Restricted Coherent Risk Measures and Actuarial Solvency

2012 ◽  
Vol 2012 ◽  
pp. 1-18
Author(s):  
Christos E. Kountzakis
Keyword(s):  
Minimization Problem ◽  
Risk Measures ◽  
Insurance Company ◽  
Coherent Risk Measures ◽  
Dual Representation ◽  
Coherent Risk ◽  
Representation Form ◽  
Solvency Capital

We prove a general dual representation form for restricted coherent risk measures, and we apply it to a minimization problem of the required solvency capital for an insurance company.

scholarly journals Digital and Blockchain-based Legal Regimes: An Eea Case Study Based on Innovative Legislations – Comparison of French and Liechtenstein Domestic Regulations

2021 ◽  
pp. 1-17
Author(s):  
Bianca Linis ◽  
Sébastien Praicheux
Keyword(s):  
Business Models ◽  
Financial Industry ◽  
Monetary System ◽  
Distributed Ledger ◽  
Distributed Ledger Technology ◽  
New Concepts ◽  
Wide Range ◽  
Extensive Range ◽  
Global Financial System

The financial crisis of 2007/08 had shattered the global financial system and led – besides a flood of regulations – to a wide range of new concepts and business models. One of these new concepts was “Bitcoin”, a private digital monetary system, which is characterized by decentralization, transparency and immutability. To date the underlying Blockchain or Distributed Ledger Technology (DLT) has evolved and offers an extensive range of possibilities, particularly in the financial industry. So far, an EU-wide legal basis for Blockchain or DLT applications and services is missing. France and the Principality of Liechtenstein took a step forward and adopted national laws trying to offer legal certainty in this field. This article aims to provide a comparison of the two acts and underline the similarities and differences.

Free-Form Modeling in Bilateral Brep and CSG Representation Schemes

1998 ◽  
Vol 08 (05n06) ◽  
pp. 537-575 ◽  
Author(s):  
Jai Menon ◽  
Baining Guo
Keyword(s):  
Shape Control ◽  
Tangent Plane ◽  
Free Form ◽  
Unified Approach ◽  
Dual Representation ◽  
Surface Patches ◽  
Low Degree ◽  
Wide Range ◽  
Algebraic Patches ◽  
Direct Term

This paper presents a unified approach for incorporating free-form solids in bilateral Brep and CSG representation schemes, by resorting to low-degree (quadratic, cubic) algebraic surface patches. We develop a general CSG solution that represents a free-form solid as a boolean combination of a direct term and a complicated delta term. This solution gives rise to the trunctet-subshell conditions, under which the delta term computation can be obviated. We use polyhedral smoothing to construct a Brep consisting of quadratic algebraic patches that meet with tangent-plane continuity, such that the trunctet-subshell conditions are guaranteed automatically. This guarantee is not currently available for cubic patches. The general CSG solution thus applies whenever trunctet-subshell conditions are violated, e.g. sometimes for cubic patches or sometimes for patches of any degree that are subject to shape control operations. Manifold solids of arbitrary topology can be represented in our dual representation system. Ensuing CSG constructs are parallel processed on the RayCasting Engine to support a wide range of solid modeling applications, including general sweeping, Minkowski operations, NC machining, and touch-sense probing.

Limit Theory for Forecasts of Extreme Distortion Risk Measures and Expectiles*

2019 ◽  
Author(s):  
Yannick Hoga
Keyword(s):  
Value At Risk ◽  
Risk Measures ◽  
Limit Theory ◽  
Extreme Risk ◽  
Wide Range ◽  
Scale Models ◽  
Adequate Coverage ◽  
Nonparametric Alternatives ◽  
Do So ◽  
Distortion Risk Measures

Abstract We develop central limit theory for tail risk forecasts in general location–scale models. We do so for a wide range of risk measures, viz. distortion risk measures (DRMs) and expectiles. Two popular members of the class of DRMs are the Value-at-Risk and the Expected Shortfall. The forecasts we consider are motivated by a Pareto-type tail assumption for the innovations and allow for extrapolation beyond the range of available observations. Simulations reveal adequate coverage of the forecast intervals derived from the limit theory. An empirical application demonstrates that our estimators outperform nonparametric alternatives when forecasting extreme risk in sufficiently large samples.

CAPITAL ALLOCATION FOR SET-VALUED RISK MEASURES

2020 ◽  
Vol 23 (01) ◽  
pp. 2050009
Author(s):  
FRANCESCA CENTRONE ◽  
EMANUELA ROSAZZA GIANIN
Keyword(s):  
Convex Set ◽  
Risk Measures ◽  
Risk Measure ◽  
Capital Allocation ◽  
Allocation Rule ◽  
Scalar Case ◽  
Representation Theorems ◽  
Allocation Rules ◽  
The One ◽  
Definition Of

We introduce the definition of set-valued capital allocation rule, in the context of set-valued risk measures. In analogy to some well known methods for the scalar case based on the idea of marginal contribution and hence on the notion of gradient and sub-gradient of a risk measure, and under some reasonable assumptions, we define some set-valued capital allocation rules relying on the representation theorems for coherent and convex set-valued risk measures and investigate their link with the notion of sub-differential for set-valued functions. We also introduce and study the set-valued analogous of some properties of classical capital allocation rules, such as the one of no undercut. Furthermore, we compare these rules with some of those mostly used for univariate (single-valued) risk measures. Examples and comparisons with the scalar case are provided at the end.

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