Abstract
A well known result in stochastic analysis reads as follows: for an \(\mathbb{R}\)-valued super-martingale X = (X t )0 ≤ t ≤ T such that the terminal value X T is non-negative, we have that the entire process X is non-negative. An analogous result holds true in the no arbitrage theory of mathematical finance: under the assumption of no arbitrage, an admissible portfolio process x + (H ⋅ S) verifying x + (H ⋅ S) T ≥ 0 also satisfies x + (H ⋅ S) t ≥ 0, for all 0 ≤ t ≤ T. In the present paper we derive an analogous result in the presence of transaction costs. In fact, we give two versions: one with a numéraire-based, and one with a numéraire-free notion of admissibility. It turns out that this distinction on the primal side perfectly corresponds to the difference between local martingales and true martingales on the dual side. A counter-example reveals that the consideration of transaction costs makes things more delicate than in the frictionless setting.
Partially supported by the Austrian Science Fund (FWF) under grant P25815, the European Research Council (ERC) under grant FA506041 and by the Vienna Science and Technology Fund (WWTF) under grant MA09-003.
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Acknowledgements
I warmly thank Irene Klein without whose encouragement this note would not have been written and who strongly contributed to its shaping. Thanks go also to Christoph Czichowsky for his advice on some of the subtle technicalities of this note. I thank an anonymous referee for careful reading and for pointing out a number of inaccuracies.
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Schachermayer, W. (2014). Admissible Trading Strategies Under Transaction Costs. In: Donati-Martin, C., Lejay, A., Rouault, A. (eds) Séminaire de Probabilités XLVI. Lecture Notes in Mathematics(), vol 2123. Springer, Cham. https://doi.org/10.1007/978-3-319-11970-0_11
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DOI: https://doi.org/10.1007/978-3-319-11970-0_11
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