Stochastic Ordering by G-Expectations
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"On Risk and Price: Stochastic Orderings and Measures "
Trans 27th ICA Werner Hürlimann (Switzerland) "On Risk And Price: Stochastic Orderings And Measures " Werner Hürlimann Switzerland Summary Following the axiomatic approach to measures of statistical quantities initiated by van Zwet(1964) and developed by several other authors, we present a general axiomatic system for the measure of the quantities risk and price. We argue that risk and insurance price are closely related through the notion of risk loading, viewed as function of the measure of risk, and that risk should be closely related to the measures of scale, skewness and kurtosis. We consider "universal" measures of scale and risk, which can be adjusted for skewness and kurtosis. Concerning the measure of price, the distortion pricing principle introduced by Denneberg(1990), studied further by Wang(1996a/b), and justified axiomatically as insurance price in a competitive market setting by Wang et al.(1997), is a measure of price for our more general axiomatic system. Our presentation includes numerous examples, some of which have so far not been encountered in actuarial science. Key words : axiomatic approach, measure of risk, measure of price, distortion pricing, stop- loss order, relative inverse convex order, scale, skewness, kurtosis Trans 27th ICA Werner Hürlimann (Switzerland) "Risque et Prix: Ordres Stochastiques et Mesures" Werner Hürlimann Switzerland Résumé. Suivant l’approche axiomatique pour les mesures de quantités statistiques initiée par van Zwet(1964) et développée par divers autres auteurs, nous présentons un système axiomatique général pour la mesure des quantités risque et prix. Nous supposons que le risque et le prix de l’assurance sont étroitement liés par la notion de surcharge de risque, conçue comme fonction de la mesure de risque, et que le risque est étroitement lié aux mesures d’échelle, d’asymétrie et d’élongation. -
Benchmark Tracking Portfolio Problems with Stochastic Ordering Constraints
Universita` degli Studi di Bergamo and VSB { Technical University of Ostrava Doctoral Thesis Benchmark Tracking Portfolio Problems with Stochastic Ordering Constraints Author: Supervisor: Marco Cassader Prof. Sergio Ortobelli Lozza Co-supervisor: doc. Ing. Tom´aˇs Tichy´, Ph.D. A thesis submitted in fulfilment of the requirements for the degree of Doctor of Philosophy in Ph.D. School in Economics, Applied Mathematics and Operational Research and Ph.D. School in Economic Policy and Administration - Finance October 2015 Declaration of Authorship I, Marco Cassader, declare that this thesis titled, \Benchmark Tracking Portfolio Problems with Stochastic Ordering Constraints" and the work presented in it are my own. I confirm that: This work was done wholly or mainly while in candidature for a research degree at these Universities. Where any part of this thesis has previously been submitted for a degree or any other qualification at these Universities or any other institution, this has been clearly stated. Where I have consulted the published work of others, this is always clearly at- tributed. Where I have quoted from the work of others, the source is always given. With the exception of such quotations, this thesis is entirely my own work. I have acknowledged all main sources of help. Where the thesis is based on work done by myself jointly with others, I have made clear exactly what was done by others and what I have contributed myself. Signed: Place and Date: Bergamo, 25/10/2015 ii \ In the end, a theory is accepted not because it is confirmed by conventional em- pirical tests, but because researchers persuade one another that the theory is correct and relevant. -
Stochastic Orders and Risk Measures: Consistency and Bounds
Stochastic Orders and Risk Measures: Consistency and Bounds Nicole B¨auerle Institut f¨urMathematische Stochastik Universit¨at Hannover, Welfengarten 1 D-30167 Hannover [email protected] Alfred M¨uller Institut f¨urWirtschaftstheorie und Operations Research Universit¨at Karlsruhe, Geb. 20.21 D-76128 Karlsruhe, Germany [email protected] Febr. 2005 Abstract We investigate the problem of consistency of risk measures with respect to usual stochastic order and convex order. It is shown that under weak regularity conditions risk measures are consistent with these stochastic orders. This result is used to derive bounds for risk measures of portfolios. As a by-product, we extend the characterization of Kusuoka (2001) of coherent, law-invariant risk measures with the Fatou property to unbounded random variables. Keywords: coherent risk measure, convex risk measure, stochastic order, convex order, copula, comonotonicity. 1 Introduction Risk measures assign a real number to a risk, which is described by a random variable X :Ω → R. As usual in actuarial sciences we assume that X describes a potential loss, but we allow X to assume negative values, which means that a gain occurs. Typically risk measures are defined by an axiomatic approach. For pioneering work in actuarial science, where risk measures occur in the context of premium principles, see e.g. Goovaerts et al. (1984). In the financial literature there is a seminal paper by Artzner et al. (1999). Reasonable axioms for risk measures which have been used throughout the literature are monotonicity, i.e. if X ≤ Y then ρ(X) ≤ ρ(Y ) and convexity, i.e. -
Extremal Moment Methods and Stochastic Orders
Boletín de la Asociación Matemática Venezolana, Vol. XV, Nro. 2 (2008) 153 Extremal Moment Methods and Stochastic Orders Application in Actuarial Science Chapters IV, V and VI Werner Hürlimann With 60 Tables Extremal Moment Methods and Stochastic Orders 155 CONTENTS PREFACE 11 CHAPTER I. Orthogonal polynomials and finite atomic random variables I.1 Orthogonal polynomials with respect to a given moment structure 13 I.2. The algebraic moment problem 17 I.3 The classical orthogonal polynomials 19 I.3.1 Hermite polynomials 19 I.3.2 Chebyshev polynomials of the first kind 20 I.3.3 Legendre polynomials 21 I.3.4 Other orthogonal polynomials 21 I.4 Moment inequalities 22 I.4.1 Structure of standard di- and triatomic random variables 22 I.4.2 Moment conditions for the existence of random variables 24 I.4.3 Other moment inequalities 26 I.5 Structure of finite atomic random variables by known moments to order four 28 I.6 Structure of finite atomic symmetric random variables by known kurtosis 33 I.7 Notes 36 CHAPTER II. Best bounds for expected values by known range, mean and variance II.1 Introduction 37 II.2 The quadratic polynomial method for piecewise linear random functions 37 II.3 Global triatomic extremal random variables for expected piecewise linear 42 transforms II.4 Inequalities of Chebyshev type 49 II.5 Best bounds for expected values of stop-loss transform type 54 II.5.1 The stop-loss transform 55 II.5.2 The limited stop-loss transform 56 II.5.3 The franchise deductible transform 57 II.5.4 The disappearing deductible transform 60 II.5.5 The two-layers stop-loss transform 62 II.6 Extremal expected values for symmetric random variables 67 II.7 Notes 73 156 Werner Hürlimann CHAPTER III.