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 Electronic Journal of Probability > Vol. 11 (2006) > Paper 3 open journal systems 


Dynamic Monetary Risk Measures for Bounded Discrete-Time Processes

Patrick Cheridito, Princeton University, USA
Freddy Delbaen, ETH Zürich, Switzerland
Michael Kupper, ETH Zürich, Switzerland


Abstract
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.


Full text: PDF

Pages: 57-106

Published on: January 26, 2006
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Electronic Journal of Probability. ISSN: 1083-6489