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Asymptotic arbitrage and numéraire portfolios in large financial markets

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Abstract

This paper deals with the notion of a large financial market and the concepts of asymptotic arbitrage and strong asymptotic arbitrage (both of the first kind) introduced in Probab. Theory Appl. 39, 222–229 (1994) and in Finance Stoch. 2, 143–172 (1998). We show that the arbitrage properties of a large market are completely determined by the asymptotic behavior of the sequence of the numéraire portfolios related to small markets. The obtained criteria can be expressed in terms of contiguity, entire separation, and Hellinger integrals, provided that these notions are extended to sub-probability measures. As examples, we consider market models on finite probability spaces, semimartingale models, and diffusion models. We also examine a discrete-time infinite horizon market model with one log-normal stock.

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Authors and Affiliations

  1. Faculty of Mathematics, Mechanics and Computer Sciences, Southern Federal University, Mil’chakova str., 8a, 344090, Rostov-on-Don, Russia

    Dmitry B. Rokhlin

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  1. Dmitry B. Rokhlin
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Correspondence to Dmitry B. Rokhlin.

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This work was supported by Southern Federal University, grant No. 26 “Mathematical Finance” and by RFBR, grant 07-01-00520.

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Rokhlin, D.B. Asymptotic arbitrage and numéraire portfolios in large financial markets. Finance Stoch 12, 173–194 (2008). https://doi.org/10.1007/s00780-007-0056-2

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