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Robust representation of convex risk measures by probability measures

Robust representation of convex risk measures by probability measures Artzner et al. [1] initiated a new direction to assess risks of financial positions by an axiomatic approach. It relies fundamentally on the concept of risk measures, which are functionals defined on sets of financial positions and satisfying some basic properties. The convex risk measures are exactly those which guarantee that diversification does not increase risk. From the standpoint of individual investors risk measures may be interpreted as loss functions expressing the preferences on the respective sets of financial positions. Starting from this point of view, Föllmer and Schied succeeded in finding a kind of robust Savage representation for convex risk measures by probability contents [3]. They also gave a sufficient condition to achieve a representation by probability measures. One aim of the paper is to show the reverse direction of their result. Another subject of the paper is to weaken the criterion within the setting when the sets of scenarios are endowed with a metrizable topology, restoring an earlier attempt by Föllmer and Schied. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Finance and Stochastics Springer Journals

Robust representation of convex risk measures by probability measures

Finance and Stochastics , Volume 9 (4) – Jan 1, 2005

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References (7)

Publisher
Springer Journals
Copyright
Copyright © 2005 by Springer-Verlag Berlin/Heidelberg
Subject
Mathematics; Quantitative Finance; Finance, general; Statistics for Business/Economics/Mathematical Finance/Insurance; Economic Theory/Quantitative Economics/Mathematical Methods; Probability Theory and Stochastic Processes
ISSN
0949-2984
eISSN
1432-1122
DOI
10.1007/s00780-005-0160-0
Publisher site
See Article on Publisher Site

Abstract

Artzner et al. [1] initiated a new direction to assess risks of financial positions by an axiomatic approach. It relies fundamentally on the concept of risk measures, which are functionals defined on sets of financial positions and satisfying some basic properties. The convex risk measures are exactly those which guarantee that diversification does not increase risk. From the standpoint of individual investors risk measures may be interpreted as loss functions expressing the preferences on the respective sets of financial positions. Starting from this point of view, Föllmer and Schied succeeded in finding a kind of robust Savage representation for convex risk measures by probability contents [3]. They also gave a sufficient condition to achieve a representation by probability measures. One aim of the paper is to show the reverse direction of their result. Another subject of the paper is to weaken the criterion within the setting when the sets of scenarios are endowed with a metrizable topology, restoring an earlier attempt by Föllmer and Schied.

Journal

Finance and StochasticsSpringer Journals

Published: Jan 1, 2005

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