Conditional and Dynamic Convex Risk Measures

We extend the definition of a convex risk measure to a conditionalframework where additional information is available. We characterize these riskmeasures through the associated acceptance sets and prove a representationresult in terms of conditional expectations. As an example we consider the classof conditional entropic risk measures. A new regularity property of conditionalrisk measures is defined and discussed. Finally we introduce the concept ofa dynamic convex risk measure as a family of successive conditional convexrisk measures and characterize those satisfying some natural time consistencyproperties.