Abstract
A consequence of the Hahn-Banach theorem is the classical bipolar theorem which states that the bipolar of a subset of a locally convex vector space equals its closed convex hull.
The space
of real-valued random variables on a probability space
equipped with the topology of convergence in measure fails to be locally convex so that—a priori—the classical bipolar theorem does not apply. In this note we show an analogue of the bipolar theorem for subsets of the positive orthant
, if we place
in duality with itself, the scalar product now taking values in [0, ∞]. In this setting the order structure of
plays an important role and we obtain that the bipolar of a subset of
equals its closed, convex and solid hull.
In the course of the proof we show a decomposition lemma for convex subsets of \(L_{^ + }^0 \left( {\Omega ,\mathcal{F},\mathbb{P}} \right)\) into a “bounded” and a “hereditarily unbounded” part, which seems interesting in its own right.
1980 Mathematics Subject Classification (1991 Revision)
- Primary: 62B20
- 28A99
- 26A20
- 52A05
- 46A55
- Secondary: 46A40
- 46N10
- 90A09
- Key words and phrases
- Convex sets of measurable functions
- Bipolar theorem
- bounded in probability
- hereditarily unbounded
The research of this paper was financially supported by the Austrian Science Foundation (FWF) under grant SFB#10 (‘Adaptive Information Systems and Modelling in Economics and Management Science’)
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References
[B 97]. W. Brannath, No Arbitrage and Martingale Measures in Option Princing, Dissertation. University of Vienna (1997).
[DS 94]. F. Delbean, W. Schachermayer, A General Version of the Fundamental Theorem of Asset Pricing, Math. Annalen 300 (1994), 463–520.
[HS 49]. Halmos, P.R., Savage, L.J. (1949), Application of the Radon-Nikodym Theorem to the Theory of Sufficient Statistics, Annals of Math. Statistics 20, 225–241.
[KS 97]. D. Kramkov, W. Schachermayer, A Condition on the Asymptotic Elasticity of Utility Functions and Optimal Investment in Incomplete Markets, Preprint (1997).
[KPR 84]. N.J. Kalton, N.T. Peck, J.W. Roberts, An F-space Sampler, London Math. Soc. Lecture Notes 89 (1984).
[M 74]. B. Maurey, Théorèmes de factorisation pour les opérateurs linéaires à valeurs dans un espace L p, Astérisque 11 (1974).
[Me 79]. P.A. Meyer, Caractérisation des semimartingales, d'après Dellacherie Séminaire de Probabilités XII, Lect. Notes Mathematics 721 (1979), 620–623.
[N 70]. E.M. Nikishin, Resonance theorems and superlinear operators, Uspekhi Mat. Nauk 25, Nr. 6 (1970), 129–191.
[S 94]. W. Schachermayer, Martingale measures for discrete time processes with infinite horizon, Math. Finance 4 (1994), 25–55.
[Sch 67]. Schaefer, H.H. (1966), Topological Vector Spaces, Springer Graduate Texts in Mathematics.
[Str 90]. Stricker, C., Arbitrage et lois de martingale, Ann. Inst. Henri. Pincaré Vol. 26, no. 3 (1990), 451–460.
[Y 80]. J. A. Yan, Caractérisation d'une classe d'ensembles convexes de L 1 ou H1, Séminaire de Probabilités XIV, Lect. Notes Mathematics 784 (1980), 220–222. *** DIRECT SUPPORT *** A00I6C60 00010
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Brannath, W., Schachermayer, W. (1999). A bipolar theorem for
.
In: Azéma, J., Émery, M., Ledoux, M., Yor, M. (eds) Séminaire de Probabilités XXXIII. Lecture Notes in Mathematics, vol 1709. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0096525
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DOI: https://doi.org/10.1007/BFb0096525
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