Stochastic Finance, 4th Edition

4.3Convex risk measures on L ∞

For the next four sections, we fix a probability measure P on (Ω,F) and consider risk measures ρ such that

In the present section, only the nullsets of P will matter.

Lemma 4.32. Let ρ be a convex risk measure that satisfies (4.33) and which is represented by a penalty function α as in (4.17). Then α(Q) = +∞ for any Q ∈ M1,f (Ω,F) which is not absolutely continuous with respect to P.

Proof. If Q ∈ M1,f (Ω,F) is not absolutely continuous with respect to P, then there exists A ∈ F such that Q[ A ] > 0 but P[ A ] = 0. Take any X ∈ Aρ, and define Xn := X − n A . Then ρ(Xn) = ρ(X), i.e., Xn is again contained in Aρ. Hence, ...