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Type of Publication: | Journal article |
Publication status: | Published |
Author: | Cheridito, Patrick; Delbaen, Freddy; Kupper, Michael |
Year of publication: | 2006 |
Published in: | Electronic Journal of Probability ; 11 (2006), 3. - pp. 57-106. - eISSN 1083-6489 |
URL of original publication: | https://projecteuclid.org/euclid.ejp/1464730538, Last access on Dec 15, 2017 |
Summary: |
We study dynamic monetary risk measures that depend on bounded discrete-time processes describing the evolution of financial values. The time horizon can be finite or infinite. We call a dynamic risk measure time-consistent if it assigns to a process of financial values the same risk irrespective of whether it is calculated directly or in two steps backwards in time. We show that this condition translates into a decomposition property for the corresponding acceptance sets, and we demonstrate how time-consistent dynamic monetary risk measures can be constructed by pasting together one-period risk measures. For conditional coherent and convex monetary risk measures, we provide dual representations of Legendre--Fenchel type based on linear functionals induced by adapted increasing processes of integrable variation. Then we give dual characterizations of time-consistency for dynamic coherent and convex monetary risk measures. To this end, we introduce a concatenation operation for adapted increasing processes of integrable variation, which generalizes the pasting of probability measures. In the coherent case, time-consistency corresponds to stability under concatenation in the dual. For dynamic convex monetary risk measures, the dual characterization of time-consistency generalizes to a condition on the family of convex conjugates of the conditional risk measures at different times. The theoretical results are applied by discussing the time-consistency of various specific examples of dynamic monetary risk measures that depend on bounded discrete-time processes.
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Subject (DDC): | 510 Mathematics |
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CHERIDITO, Patrick, Freddy DELBAEN, Michael KUPPER, 2006. Dynamic Monetary Risk Measures for Bounded Discrete-Time Processes. In: Electronic Journal of Probability. 11(3), pp. 57-106. eISSN 1083-6489
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