lattice. For the space of bounded positions from X the symbol Xb will be used. Furthermore let us fix a vector
subspace C ⊆ X of financial positions for hedging, including the constants. In particular we may take into account
liquid derivatives like put and call options as financial instruments. They are associated with a positive linear
function π : C → R, π(1) = 1, where π(Y ) stands for the initial costs to obtain Y. Prominent special cases of this
setting are the following:
• T = {1,...,n}, (Ft)t∈T family of σ-algebras on a set of scenarios Ω, X consisting of all X ∈ Rω×t with
X(∙,t) Ft - measurable for t ∈ T, C = Rn, π(yι,...,yn):= 1 PP yi
n i=1
n-period positions, one-period positions if n = 1
• T = [0,T], (Ft)t∈T be a filtration of σ-algebras on a set of scenarios Ω, X set of financial positions X with
X(ω, ∙) being a cadlag function, C = R, π identity on R
cadlag positions
Let us now introduce the concept of risk measures suggested by Frittelli and Scandolo in [9]. As for one-period
positions we may choose the axiomatic viewpoint, defining a risk measure w.r.t. π to be a functional ρ : X → R
which satisfies the properties
• monotonicity: ρ(X) ≤ ρ(Y ) for X ≥ Y
• translation invariance w.r.t. π: ρ(X + Y ) = ρ(X) - π(Y ) for X ∈ X, Y ∈ C
The meaning of these conditions may be transferred from the genuine concept of risk measures. Moreover, it can
be shown that a risk measure ρ w.r.t. π satisfies ρ(X) = inf {π(Y ) | Y ∈ C, ρ(X + Y ) ≤ 0} for any X ∈ X ([9],
Proposition 3.6). Regarding ρ-1(] - ∞, 0]) as the acceptable positions, an outcome ρ(X) expresses the infimal
costs to hedge it. This retains the original meaning of risk measures as capital requirements.
In the following we shall focus on so-called convex risk measures, defined to mean risk measures which are convex
mappings. Convexity is a reasonable condition for a risk measure due to its interpretation that diversification
should not increase risk. From the technical point of view convexity is a necessary property for the desired dual
representations of risk measures.
Let us now fix a convex risk measure ρ : X → R w.r.t. π. It is associated with the so-called Fenchel-Legendre
transform
βρ : X* →] — ∞, ∞], Λ → sup (—Λ(X) — ρ(X)),
X∈X
where X* gathers all real linear forms on X. It is easy to verify that every Λ from the domain of βρ has to be a
positive linear form extending π. The standard tools from convex analysis provide basic representation results for
ρ with βρ as a penalty function.
Proposition 1.1 Let X*+π denote the space of all positive linear forms on X which extend π, and let τ be any
topology on X such that (X,τ) is a locally convex topological vector space with topological dual X. Then ρ(X) =