On s-additive robust representation of convex risk measures for unbounded financial positions in the presence of uncertainty about the market model

Therefore Λ := — Λ is a positive linear form fulfilling Λ(X — Y) ≤ ρ+(X, Y — X) ≤ ρ(Y) — ρ(X) for Y ∈ X,

and Λ(Y) ≤ π(Y) for Y ∈ C. This implies Λ∣C = π due to linearity of Λ∣C and π. Furthermore we have shown

βρ(Λ) = —Λ(X ) — ρ(X ).

For the proof of the equivalence stated in Proposition 1.1 note that the only if part is obvious, the if part follows
immediately from the Fenchel-Moreau theorem (cf. [2], Theorem 4.2.2), observing that βρ (Λ) < ∞ only if Λ is
positive linear and extends π (see also Proposition 3.9 in [9]). This completes the proof.                       H

Acknowlegdements:

The author would like to thank Freddy Delbaen and Alexander Schied for helpful hints. He is also indepted to
Heinz Konig for good advice and continuous exchange of ideas from the fields of measure theory and superconvex
analysis.

References

[1] P. Artzner, F. Delbaen, J.-M. Eber, D. Heath, Coherent measures of risk, Math. Finance 9 (1999), 203-228.

[2] J.-P. Aubin, I. Ekeland, ”Applied Nonlinear Analysis”, John Wiley and Sons, New York, 1984.

[3] C. Castaing, M. Valadier, ”Convex Analysis and Measurable Multifunctions”, Lecture Notes in Mathematics
Vol. 580, Springer, Berlin, Heidelberg, New York, 1977.

[4] F. Delbaen, Coherent Measures of Risk on General Probability Spaces, in: K. Sandmann, PJ. Schonbucher
(Eds.), ”Advances in Finance and Stochastics”, Springer, Berlin, 2002, pp. 1-37.

[5] C. Dellacherie, P.-A. Meyer, ”Probabilities and Potential”, North-Holland, Amsterdam, 1978.

[6] D. Denneberg, ”Non-Additive Measure and Integral”, Kluwer, Dordrecht, 1994.

[7] H. Follmer, A. Schied, ’’Stochastic Finance”, de Gruyter, Berlin, New York, 2004 (2nd ed.).

[8] M. Fritelli, E. Rosazza Gianin, Dynamic Convex Risk Measures, in: G. Szego (Ed), ”Risk Measures for the
21st Century”, Wiley, New York, 2004, pp. 227-248.

[9] M. Frittelli, G. Scandolo, Risk measures and capital requirements for processes, Math. Finance 16 (2006),
589-612.

[10] B. Fuchssteiner, W. Lusky, ”Convex Cones”, North-Holland, Amsterdam, New York, 1981.

[11] E. Jouini, W. Schachermayer and N. Touzi, Law invariant risk measures have the Fatou property, Advances
in Mathematical Economics, 9 (2006), 49-71.

[12] H. Konig, On some basic theorems in convex analysis, in: B. Korte (Ed.), ’Modern Applied Mathematics -
Optimization and Operations Research”, North-Holland, Amsterdam,1982, pp. 107-144.

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