Representation and approximation of convex dynamic risk measures with respect to strong-weak topologies

Okhrati, Ramin and Assa, Hirbod (2017) Representation and approximation of convex dynamic risk measures with respect to strong-weak topologies. Stochastic Analysis and Applications, 35 (4), 1-11. (doi:10.1080/07362994.2017.1289104).

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Abstract

We provide a representation for strong-weak continuous dynamic risk measures from Lp into Lpt 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 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.

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