Vigilant measures of risk and the demand for contingent claims |
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| Affiliation: | 1. Fachbereich Mathematik, Technische Universität Kaiserslautern, Erwin-Schrödinger Straße, 67653 Kaiserslautern, Germany;2. Fachgruppe Stochastik am Mathematischen Seminar, Christian-Albrechts-Universität zu Kiel, Ludewig-Meyn-Straße 4, 24098 Kiel, Germany;3. Department of Mathematics, SPST, University of Hamburg, Bundesstrasse 55, 20146 Hamburg, Germany;4. School of Mathematical Sciences, Dublin City University, Dublin 9, Ireland;1. Department of Statistics and Finance, University of Science and Technology of China, Hefei, Anhui 230026, China;2. Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, ON N2L3G1, Canada;1. Carlos III University of Madrid, C/Madrid, 126, 28903 Getafe, Spain;2. Sorbonnes Université, Université de Technologie de Compiègne, CNRS, UMR Heudiasyc, 57 Av. de Landshut, 60203 Compiègne, France;1. Faculty of Science, Huzhou University, Huzhou, Zhejiang 313000, China;2. School of Sciences, Communication University of China, Beijing 100024, China;3. Faculty of Civil Engineering, Slovak University of Technology, Radlinského 11, 813 68 Bratislava, Slovakia;4. UTIA CAS, Pod Vodárenskou věží 4, 182 08 Prague, Czech Republic |
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| Abstract: | We examine a class of utility maximization problems with a non-necessarily law-invariant utility, and with a non-necessarily law-invariant risk measure constraint. Under a consistency requirement on the risk measure that we call Vigilance, we show the existence of optimal contingent claims, and we show that such optimal contingent claims exhibit a desired monotonicity property. Vigilance is satisfied by a large class of risk measures, including all distortion risk measures and some classes of robust risk measures. As an illustration, we consider a problem of optimal insurance design where the premium principle satisfies the vigilance property, hence covering a large collection of commonly used premium principles, including premium principles that are not law-invariant. We show the existence of optimal indemnity schedules, and we show that optimal indemnity schedules are nondecreasing functions of the insurable loss. |
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| Keywords: | Utility maximization Optimal insurance design Choquet integral Distorted probabilities Monotone Likelihood Ratio |
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