Representation and approximation of convex dynamic risk measures with respect to strong–weak topologies
Stochastic Analysis and Applications
We provide a representation for strong-weak continuous dynamic risk measures from L p into L p t spaces where these spaces are equipped respectively with strong and weak topologies and p is a finite number strictly larger than one. Conversely, we show that any such representation that admits a compact (with respect to the product of weak topologies) sub-differential generates a dynamic risk measure that is strong-weak continuous. Furthermore, we investigate sufficient conditions on the
... ons on the sub-differential for which the essential supremum of the representation is attained. Finally, the main purpose is to show that any convex dynamic risk measure that is strong-weak continuous can be approximated by a sequence of convex dynamic risk measures which are strong-weak continuous and admit compact sub-differentials with respect to the product of weak topologies. Throughout the arguments, no conditional translation invariance or monotonicity assumptions are applied.