Conditional risk and acceptability mappings as Banach-lattice valued mappings

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 L_p(Ω,F,ℙ) → L_p´(Ω,F´,ℙ) with 1 ≤ p´ ≤ p ≤ ∞, where the σ-algebras F´ ⊂ F describe the available information. Based on the order completeness of L_p(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.