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**Conditional and dynamic convex risk measures.**
*(English)*
Zbl 1092.91017

The authors extend the definition of a convex risk measure to a conditional framework where additional information is available. Conditional convex/coherent risk measures are introduced and the property of regularity is discussed. It is proved that every conditional convex risk measure can be interpreted as a conditional capital requirement. A characterization result is formulated in terms of conditional expectations and some minor results about the random penalty functions are formulated. The conditional entropy risk measures are defined, providing an explicit dual representation. Then, dynamic convex risk measures are introduced as a family of successive conditional convex risk measures and three natural time-consistency properties are discussed. As a reference example, a suitably defined dynamic version of the class of entropy risk measures is considered.

Reviewer: Yuliya Mishura (Kyïv)

### MSC:

91B16 | Utility theory |

91B30 | Risk theory, insurance (MSC2010) |

91B70 | Stochastic models in economics |

46A20 | Duality theory for topological vector spaces |