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May 2021 Large deviations built on max-stability
Michael Kupper, José Miguel Zapata
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Bernoulli 27(2): 1001-1027 (May 2021). DOI: 10.3150/20-BEJ1263

Abstract

In this paper, we show that the basic results in large deviations theory hold for general monetary risk measures, which satisfy the crucial property of max-stability. A max-stable monetary risk measure fulfills a lattice homomorphism property, and satisfies under a suitable tightness condition the Laplace Principle (LP), that is, admits a dual representation with affine convex conjugate. By replacing asymptotic concentration of probability by concentration of risk, we formulate a Large Deviation Principle (LDP) for max-stable monetary risk measures, and show its equivalence to the LP. In particular, the special case of the asymptotic entropic risk measure corresponds to the classical Varadhan–Bryc equivalence between the LDP and LP. The main results are illustrated by the asymptotic shortfall risk measure.

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Michael Kupper. José Miguel Zapata. "Large deviations built on max-stability." Bernoulli 27 (2) 1001 - 1027, May 2021. https://doi.org/10.3150/20-BEJ1263

Information

Received: 1 April 2020; Revised: 1 July 2020; Published: May 2021
First available in Project Euclid: 24 March 2021

Digital Object Identifier: 10.3150/20-BEJ1263

Keywords: asymptotic shortfall risk , concentration function , Laplace principle , large deviation principle , large deviations , max-stable monetary risk measures

Rights: Copyright © 2021 ISI/BS

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Vol.27 • No. 2 • May 2021
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