Representation and approximation of convex dynamic risk measures with respect to strong–weak topologies

ABSTRACT

We provide a representation for strong-weak continuous dynamic risk measures from L^p into L^p_t spaces where these spaces are equipped respectively with strong and weak topologies and p is a finite number strictly larger than one. Conversely, we show that any such representation that admits a compact (with respect to the product of weak topologies) sub-differential generates a dynamic risk measure that is strong--weak continuous. Furthermore, we investigate sufficient conditions on the sub-differential for which the essential supremum of the representation is attained. Finally, the main purpose is to show that any convex dynamic risk measure that is strong-weak continuous can be approximated by a sequence of convex dynamic risk measures which are strong--weak continuous and admit compact sub-differentials with respect to the product of weak topologies. Throughout the arguments, no conditional translation invariance or monotonicity assumptions are applied.

KEYWORDS:

• Convexity
• dynamic risk measures
• representation theorem
• sub-differential
• weak and strong continuity

MATHEMATICS SUBJECT CLASSIFICATION:

• 91G99
• 60H30
• 46N10

Acknowledgments

R. Okhrati is very thankful to Alejandro Balbás for helping with the proof of Lemma A.1. Both authors are grateful to an anonymous referee for constructive comments.

Notes

1. The terminology weak-weak means that domain and the range of the risk measure are equipped with weak topologies. The other terminologies such as strong–weak or strong–strong are defined in the same fashion.

2. Here, the convexity means that if (z₁, k₁) ∈ Δ and (z₂, k₂) ∈ Δ then (λ(z₁, k₁) + (1 − λ)(z₂, k₂)) ∈ Δ for all 0 ⩽ λ ⩽ 1, .

3. This can be proved using Hahan–Banach separating hyperplanes theorem.

4. It can be proved that the continuity of a proper convex function on normed spaces is equivalent to its continuity at origin.

5. Note that if the image space of ρ_t is a subspace of real numbers then this is rather a trivial result from functional analysis facts.

6. Note that this is true whether or not the set is directed upward.