Conditional risk and acceptability mappings as Banach-lattice valued mappings

Raimund Kovacevic ()

Statistics & Risk Modeling, 2012, vol. 29, issue 1, 1-18

Abstract: Conditional risk and acceptability mappings quantify the desirability of random variables (e.g. financial returns) by accounting for available information. In this paper the focus lies on acceptability mappings, concave translation-equivariant monotone mappings Lp(Ω,F,ℙ) → Lp´(Ω,F´,ℙ) with 1 ≤ p´ ≤ p ≤ ∞, where the σ-algebras F´ ⊂ F describe the available information. Based on the order completeness of Lp(Ω,F,ℙ)-spaces, we analyze superdifferentials and concave conjugates of conditional acceptability mappings. The related results are used to show properties of two important classes of multi-period valuation functionals: SEC-functionals and additive acceptability compositions. In particular, we derive a chain rule for superdifferentials and use it for characterizing the conjugates of additive acceptability compositions and SEC-functionals.

Keywords: conditional risk measure; conditional acceptability mapping; Banach lattices; convex analysis (search for similar items in EconPapers)
Date: 2012
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DOI: 10.1524/strm.2012.1041

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