Stochastic Finance, 4th Edition

4.2Robust representation of convex risk measures

In this section, we consider a situation of Knightian uncertainty, where no probability measure P is fixed on the measurable space (Ω,F). Let X denote the space of all bounded measurable functions on (Ω,F). Recall that X is a Banach space if endowed with the supremum norm · . As in Section 2.5, we denote by M1 := M1(Ω,F) the set of all probability measures on (Ω,F) and by M1,f := M1,f (Ω,F) the set of all finitely additive set functions Q : F → [0, 1] which are normalized to Q[ Ω ] = 1. By EQ[ X ] we denote the integral of X with respect to Q ∈ M1,f ; see Appendix A.6.

If ρ is a coherent risk measure on X , thenwe are in the context of Proposition 2.84, i.e., the functional ϕ defined by ϕ(X) := ...