Nonsmooth Mechanics and Analysis
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Risk, Uncertainty and Optimisation on Infinite Dimensional Ordered Spaces
Risk, Uncertainty and Optimisation on Infinite Dimensional Ordered Spaces Dissertation an der Fakult¨atf¨urMathematik, Informatik und Statistik der Ludwig-Maximilians-Universit¨atM¨unchen vorgelegt von Felix-Benedikt Liebrich 7. August 2019 1. Gutachter: Priv.-Doz. Dr. Gregor Svindland 2. Gutachter: Prof. Dr. Michael Kupper 3. Gutachterin: Prof. Dr. Francesca Biagini Tag der m¨undlichen Pr¨ufung:19. Februar 2020 Eidesstattliche Versicherung (Siehe Promotionsordnung vom 12.07.11, § 8, Abs. 2 Pkt. 5.) Ich versichere hiermit an Eidesstatt, dass die vorgelegte Dissertation von mir selbst¨andig,ohne unerlaubte Beihilfe angefertigt ist. M¨unchen, den 7. August 2019 Felix-Benedikt Liebrich Zusammenfassung Die vorliegende Arbeit beleuchtet mathematische Aspekte der Theorie ¨okonomischer und fi- nanzieller Risiken. In den drei versammelten Beitr¨agenwerden das Fortsetzungsproblem f¨ur Risikomaße unter Knight’scher Unsicherheit sowie die Optimierung von Risiko- und Nutzenauf- teilungen studiert. Der erste Beitrag, Model Spaces for Risk Measures, widmet sich dem Fortsetzungsproblem f¨ur Risikomaße. Letztere sind als Kapitalanforderungen definiert, welche durch das Zusammenspiel einer konvexen Menge akzeptabler Positionen und eines mehrdimensionalen Security-Markts zur Absicherung nicht akzeptabler Positionen induziert werden. Die Nettoverluste, deren Risiko gemessen wird, werden durch Zufallsvariablen auf der Menge m¨oglicher k¨unftigerZust¨andeder Okonomie¨ modelliert. Wir betrachten allerdings Risikomaße, welche auch Knight’sche Unsicher- heit abbilden: -
Well-Positioned Convex Sets and Functions and Applications Michel Thera´ LACO, UMR-CNRS 6090, University of Limoges [email protected]
Well-positioned convex sets and functions and applications Michel Thera´ LACO, UMR-CNRS 6090, University of Limoges [email protected] Main collaborators : S. Adly, E. Ernst and C. Zalinescu Naples, december 19 2005 p.1/?? Credits E. ERNST and M. THERA´ ,C.ZALINESCU˘ , Constrained optimization and strict convex separation, Slice-continuous sets in reflexive Banach spaces: convex constrained optimization and strict convex separation, Journal of Functional Analysis, Vol 223/1 pp 179-203, 2005. S. ADLY,E.ERNST and M. THERA´ , Well-positioned closed convex sets and well-positioned closed convex functions, Journal of Global Optimization 29 (4): 337-351, (2004). E. ERNST and M. THERA´ , Continuous sets and non-attaining functionals in reflexive Banach spaces, Variational Analysis and Applications, F. Giannessi and A. Maugeri, Eds, Springer, 2005 in press. p.2/?? Credits S. ADLY,E.ERNST and M. THERA´ , Stability of Non-coercive Variational Inequalities, Communications in Contemporary Mathematics, Vol 4, 1, 145 – 160, 2002. S. ADLY,E.ERNST and M. THERA´ , On the converse of the Dieudonné Theorem in Reflexive Banach Spaces, Special volume, dedicated to the memory of Boris Pshenichnyi, Cybernetics & System Analysis, No 3, 34 – 39, 2002. S. ADLY,E.ERNST and M. THERA´ , On the closedness of the algebraic difference of closed convex sets, Journal de Mathématiques Pures et Appliquées, Volume 82, No 9, 1219 – 1249, 2003. p.3/?? Outline Setting p.4/?? Outline Setting Some properties of the barrier cone to a convex set p.5/?? Outline Setting Some properties -
Addition and Subtraction of Homothety Classes of Convex Sets
Beitr¨agezur Algebra und Geometrie Contributions to Algebra and Geometry Volume 47 (2006), No. 2, 351-361. Addition and Subtraction of Homothety Classes of Convex Sets Valeriu Soltan Department of Mathematical Sciences, George Mason University 4400 University Drive, Fairfax, VA 22030, USA e-mail: [email protected] Abstract. Let SH denote the homothety class generated by a convex n n set S ⊂ R : SH = {a + λS | a ∈ R , λ > 0}. We determine conditions for the Minkowski sum BH + CH or the Minkowski difference BH ∼ CH of homothety classes BH and CH generated by closed convex sets B, C ⊂ Rn to lie in a homothety class generated by a closed convex set (more generally, in the union of countably many homothety classes generated by closed convex sets). MSC 2000: 52A20 Keywords: convex set, homothety class, Minkowski sum, Minkowski difference 1. Introduction and main results In what follows, everything takes place in the Euclidean space Rn. Let us recall that a set B is homothetic to a set A provided B = a + λA for a suitable point a and a scalar λ > 0. If A is a convex set, then the Minkowski sum of any two homothetic copies of A is again a homothetic copy of A. In other words, the homothety class n AH = {a + λA | a ∈ R , λ > 0} is closed with respect to the Minkowski addition. We will say that closed convex sets B and C form a pair of H-summands of a closed convex set A, or summands of A with respect to homotheties, provided the Minkowski sum of any homothetic copies of B and C is always homothetic to A. -
Risk Sharing for Capital Requirements with Multidimensional Security Markets
RISK SHARING FOR CAPITAL REQUIREMENTS WITH MULTIDIMENSIONAL SECURITY MARKETS FELIX-BENEDIKT LIEBRICH GREGOR SVINDLAND Department of Mathematics, University of Munich, Germany Abstract. We consider the risk sharing problem for capital requirements induced by cap- ital adequacy tests and security markets. The agents involved in the sharing procedure may be heterogeneous in that they apply varying capital adequacy tests and have access to different security markets. We discuss conditions under which there exists a representative agent. Thereafter, we study two frameworks of capital adequacy more closely, polyhedral constraints and distribution based constraints. We prove existence of optimal risk alloca- tions and equilibria within these frameworks and elaborate on their robustness. Keywords: capital requirements, polyhedral acceptance sets, law-invariant acceptance sets, multidimensional security spaces, Pareto optimal risk allocations, equilibria, robustness of optimal allocations. E-mail address: [email protected], [email protected]. Date: September 25th, 2018. 2010 Mathematics Subject Classification. 91B16, 91B30, 91B32, 91B50. 1 2 1. Introduction In this paper we consider the risk sharing problem for capital requirements. Optimal cap- ital and risk allocation among economic agents, or business units, has for decades been a predominant subject in the respective academic and industrial research areas. Measuring financial risks with capital requirements goes back to the seminal paper by Artzner et al. [5]. There, risk measures are by definition capital requirements determined by two primitives: the acceptance set and the security market. The acceptance set, a subset of an ambient space of losses, corresponds to a capital adequacy test. A loss is deemed adequately capitalised if it belongs to the acceptance set, and inad- equately capitalised otherwise. -
Risk Sharing with Multidimensional Security Markets
RISK SHARING WITH MULTIDIMENSIONAL SECURITY MARKETS FELIX-BENEDIKT LIEBRICH GREGOR SVINDLAND Department of Mathematics, University of Munich, Germany Abstract. We consider the risk sharing problem for capital requirements induced by cap- ital adequacy tests and security markets. The agents involved in the sharing procedure may be heterogeneous in that they apply varying capital adequacy tests and have access to different security markets. We discuss conditions under which there exists a representative agent. Thereafter, we study two frameworks of capital adequacy more closely, polyhedral constraints and distribution based constraints. We prove existence of optimal risk alloca- tions and equilibria within these frameworks and elaborate on their robustness. Keywords: capital requirements, polyhedral acceptance sets, law-invariant acceptance sets, multidimensional security spaces, Pareto-optimal risk allocations, equilibria, robustness of optimal allocations. JEL Classification: D52, D53, G12 2010 Mathematics Subject Classification: 91B16, 91B30, 91B32, 91B50 E-mail address: [email protected], [email protected]. Date: August 2019. 1 2 1. Introduction In this paper we consider the risk sharing problem for capital requirements. Optimal cap- ital and risk allocation among economic agents, or business units, has for decades been a predominant subject in the respective academic and industrial research areas. Measuring financial risks with capital requirements goes back to the seminal paper by Artzner et al. [6]. There, risk measures are by definition capital requirements determined by two primitives: the acceptance set and the security market. The acceptance set, a subset of an ambient space of losses, corresponds to a capital adequacy test. A loss is deemed adequately capitalised if it belongs to the acceptance set, and inad- equately capitalised otherwise. -
Convex Analysis1
Convex Analysis1 Patrick Cheridito Princeton University, Spring 2013 1Many thanks to Andreas Hamel for providing his lecture notes. Large parts of chapters 2-4 are based on them. 2 Contents Contents 2 1 Convex Analysis in Rd 5 1.1 Subspaces, affine sets, convex sets, cones and half-spaces . 5 1.2 Separation results in finite dimensions . 8 1.3 Linear, affine and convex functions . 12 1.4 Derivatives, directional derivatives and sub-gradients . 16 1.5 Convex conjugates . 19 1.6 Inf-convolution . 23 2 General Vector Spaces 27 2.1 Definitions . 27 2.2 Zorn's lemma and extension results . 30 2.3 Algebraic interior and separation results . 33 2.4 Directional derivatives and sub-gradients . 35 3 Topological Vector Spaces 37 3.1 Topological spaces . 37 3.2 Continuous linear functionals and extension results . 42 3.3 Separation with continuous linear functionals . 44 3.4 Continuity of convex functions . 47 3.5 Derivatives and sub-gradients . 48 3.6 Dual pairs . 50 3.7 Convex conjugates . 53 3.8 Inf-convolution . 55 4 Convex Optimization 59 4.1 Perturbation and the dual problem . 59 4.2 Lagrangians and saddle points . 65 4.3 Karush{Kuhn{Tucker-type conditions . 67 3 4 CONTENTS Chapter 1 d Convex Analysis in R The following notation is used: • d 2 N := f1; 2;:::g d • ei is the i-th unit vector in R Pd d • hx; yi := i=1 xiyi for x; y 2 R p • jjxjj := hx; xi for x 2 Rd d • B"(x) := y 2 R : jjx − yjj ≤ " • R+ := fx 2 R : x ≥ 0g, R++ := fx 2 R : x > 0g • x _ y := max fx; yg and x ^ y := min fx; yg for x; y 2 R 1.1 Subspaces, affine sets, convex sets, cones and half-spaces Definition 1.1.1 Let C be a subset of Rd.