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Law-invariant functionals on general spaces of random variables
TL;DR: In this paper, the equivalence of law invariance and Schur convexity for law-invariant functions is established for a large class of spaces of random variables.
Abstract: We establish general versions of a variety of results for quasiconvex, lower-semicontinuous, and law-invariant functionals. Our results extend well-known results from the literature to a large class of spaces of random variables. We sometimes obtain sharper versions, even for the well-studied case of bounded random variables. Our approach builds on two fundamental structural results for law-invariant functionals: the equivalence of law invariance and Schur convexity, i.e., monotonicity with respect to the convex stochastic order, and the fact that a law-invariant functional is fully determined by its behaviour on bounded random variables. We show how to apply these results to provide a unifying perspective on the literature on law-invariant functionals, with special emphasis on quantile-based representations, including Kusuoka representations, dilatation monotonicity, and infimal convolutions.
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TL;DR: In this paper, it was shown that the expectation functional is the only law-invariant convex functional that is linear along the direction of a nonconstant random variable with nonzero expectation.
Abstract: We discuss when law-invariant convex functionals "collapse to the mean". More precisely, we show that, in a large class of spaces of random variables and under mild semicontinuity assumptions, the expectation functional is, up to an affine transformation, the only law-invariant convex functional that is linear along the direction of a nonconstant random variable with nonzero expectation. This extends results obtained in the literature in a bounded setting and under additional assumptions on the functionals. We illustrate the implications of our general results for pricing rules and risk measures.
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TL;DR: In this paper, the authors present and justify a set of four desirable properties for measures of risk, and call the measures satisfying these properties "coherent", and demonstrate the universality of scenario-based methods for providing coherent measures.
Abstract: In this paper we study both market risks and nonmarket risks, without complete markets assumption, and discuss methods of measurement of these risks. We present and justify a set of four desirable properties for measures of risk, and call the measures satisfying these properties “coherent.” We examine the measures of risk provided and the related actions required by SPAN, by the SEC=NASD rules, and by quantile-based methods. We demonstrate the universality of scenario-based methods for providing coherent measures. We offer suggestions concerning the SEC method. We also suggest a method to repair the failure of subadditivity of quantile-based methods.
8,651 citations
Book•
01 Jan 1976TL;DR: In this article, the authors consider non-convex variational problems with a priori estimate in convex programming and show that they can be solved by the minimax theorem.
Abstract: Preface to the classics edition Preface Part I. Fundamentals of Convex Analysis. I. Convex functions 2. Minimization of convex functions and variational inequalities 3. Duality in convex optimization Part II. Duality and Convex Variational Problems. 4. Applications of duality to the calculus of variations (I) 5. Applications of duality to the calculus of variations (II) 6. Duality by the minimax theorem 7. Other applications of duality Part III. Relaxation and Non-Convex Variational Problems. 8. Existence of solutions for variational problems 9. Relaxation of non-convex variational problems (I) 10. Relaxation of non-convex variational problems (II) Appendix I. An a priori estimate in non-convex programming Appendix II. Non-convex optimization problems depending on a parameter Comments Bibliography Index.
4,434 citations
Book•
01 Jan 1987
TL;DR: In this article, the classical interpolation theorem is extended to the Banach Function Spaces, and the K-Method is used to find a Banach function space with a constant number of operators.
Abstract: Banach Function Spaces. Rearrangement-Invariant Banach Function Spaces. Interpolation of Operators on Rearrangement-Invariant Spaces. The Classical Interpolation Theorems. The K-Method. Each chapter includes references. Index.
3,388 citations
Book•
12 Mar 2014
TL;DR: In this paper, Riesz spaces are used to represent the topology of the space of sequences of sequences and correspondences of correspondences in Markov transitions, where the correspondences correspond to Markov transition.
Abstract: Odds and ends- Topology- Metrizable spaces- Measurability- Topological vector spaces- Normed spaces- Convexity- Riesz spaces- Banach lattices- Charges and measures- Integrals- Measures and topology- Lp-spaces- Riesz Representation Theorems- Probability measures- Spaces of sequences- Correspondences- Measurable correspondences- Markov transitions- Ergodicity
2,221 citations
Book•
01 Jan 2002
TL;DR: In this article, the authors present an introduction to financial mathematics, focusing on stochastic models in discrete time, with a focus on the problem of pricing and hedging of financial derivatives.
Abstract: This book is an introduction to financial mathematics. It is intended for graduate students in mathematics and for researchers working in academia and industry. The focus on stochastic models in discrete time has two immediate benefits. First, the probabilistic machinery is simpler, and one can discuss right away some of the key problems in the theory of pricing and hedging of financial derivatives. Second, the paradigm of a complete financial market, where all derivatives admit a perfect hedge, becomes the exception rather than the rule. Thus, the need to confront the intrinsic risks arising from market incomleteness appears at a very early stage. The first part of the book contains a study of a simple one-period model, which also serves as a building block for later developments. Topics include the characterization of arbitrage-free markets, preferences on asset profiles, an introduction to equilibrium analysis, and monetary measures of financial risk. In the second part, the idea of dynamic hedging of contingent claims is developed in a multiperiod framework. Topics include martingale measures, pricing formulas for derivatives, American options, superhedging, and hedging strategies with minimal shortfall risk. This third revised and extended edition now contains more than one hundred exercises. It also includes new material on risk measures and the related issue of model uncertainty, in particular a new chapter on dynamic risk measures and new sections on robust utility maximization and on efficient hedging with convex risk measures.
1,866 citations