Abstract.
We extend the definition of a convex risk measure to a conditional framework where additional information is available. We characterize these risk measures through the associated acceptance sets and prove a representation result in terms of conditional expectations. A suitable regularity property of conditional risk measures is defined and discussed. Finally, we introduce the concept of a dynamic convex risk measure as a family of successive conditional convex risk measures and characterize those satisfying some natural time consistency properties. As a reference example, illustrating all the proposed developments, we introduce a suitably defined dynamic version of the class of entropic risk measures.
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Mathematics Subject Classification:
91B16, 91B70, 91B30, 46A20
JEL Classification:
D81
The authors wish to thank Hans Föllmer, Marco Frittelli and Alexander Schied for useful discussions and an anonymous referee for valuable suggestions about the literature.
Manuscript received: August 2004; final version received: March 2005
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Detlefsen, K., Scandolo, G. Conditional and dynamic convex risk measures. Finance Stochast. 9, 539–561 (2005). https://doi.org/10.1007/s00780-005-0159-6
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DOI: https://doi.org/10.1007/s00780-005-0159-6