Representation and approximation of convex dynamic risk measures with respect to strong–weak topologies. Issue 4 (4th July 2017)

• Record Type: Journal Article
• Authors: Okhrati, Ramin; Assa, Hirbod
• Is Part Of: Stochastic analysis and applications. Volume 35:Issue 4(2017)
• Page Start: 604
• Page End: 614
• Publication Date: 2017-07-04
• Subjects: Convexity -- dynamic risk measures -- representation theorem -- sub-differential -- weak and strong continuity; 91G99 -- 60H30 -- 46N10; Stochastic analysis -- Periodicals; 519.2205
• DOI: 10.1080/07362994.2017.1289104
• Languages: English
• ISSNs: 0736-2994

Abstract

ABSTRACT: 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 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.