Extremal Moment Methods and Stochastic Orders
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Stochastic Ordering by G-Expectations
Stochastic ordering by g-expectations Sel Ly Nicolas Privault School of Physical and Mathematical Sciences Division of Mathematical Sciences Nanyang Technological University Singapore 637371 May 27, 2020 Abstract We derive sufficient conditions for the convex and monotonic g-stochastic ordering of diffusion processes under nonlinear g-expectations and g-evaluations. Our approach relies on comparison results for forward-backward stochastic differential equations and on several extensions of convexity, monotonicity and continuous dependence proper- ties for the solutions of associated semilinear parabolic partial differential equations. Applications to contingent claim price comparison under different hedging portfolio constraints are provided. Keywords: Stochastic ordering, g-expectation, g-evaluation, g-risk measures, forward-backward stochastic differential equations, parabolic PDEs, propagation of convexity. Mathematics Subject Classification: 60E15; 35B51; 60H10; 60H30. 1 Introduction In comparison with standard mean-variance analysis, partial orderings of probability distri- butions provide additional information which can be used in applications to risk management. arXiv:2005.12580v1 [math.PR] 26 May 2020 In this framework, a random variable X is said to be dominated by another random variable Y if E[φ(X)] ≤ E[φ(Y )], (1.1) for all φ : R → R in a certain class of functions, where E[X] denotes the usual expectation of the random variable X. For example, if X and Y represent the lifetimes of two devices A and B then the stochastic ordering (1.1) for all non-decreasing and bounded functions φ, tells that the device B will be likely survive longer than the device A. More generally, stochastic ordering has found numerous applications in fields such as reliability, economics, actuarial 1 science, comparison of experiments etc, see e.g. -
"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.