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Exponentially concave functions and high dimensional stochastic portfolio theory

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  • Pal, Soumik

Abstract

We construct an explicit example of asymptotic short term relative arbitrage. Specifically, for every n we assume an n dimensional semimartingale market model that starts from a heavy-tailed initial position in the unit simplex and impose weak assumptions on its volatility. We then construct a sequence of portfolios, one for each dimension, that outperform the market portfolio in dimension n by an amount Mn by time δn with a probability at least 1−qn. Here Mn→∞ exponentially fast in n and δn,qn decrease to zero. Moreover, these portfolios never underperform below a pre-specified lower bound. The key fact is that it is possible to construct a sequence of exponentially concave functions on the unit simplex of increasing concavity because the typical diameter of the unit simplex in dimension n is O(1∕n).

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  • Pal, Soumik, 2019. "Exponentially concave functions and high dimensional stochastic portfolio theory," Stochastic Processes and their Applications, Elsevier, vol. 129(9), pages 3116-3128.
    • Handle: RePEc:eee:spapps:v:129:y:2019:i:9:p:3116-3128
    DOI: 10.1016/j.spa.2018.09.004
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    References listed on IDEAS

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    Cited by:

    1. Martin Larsson & Johannes Ruf, 2020. "Relative Arbitrage: Sharp Time Horizons and Motion by Curvature," Papers 2003.13601, arXiv.org, revised Feb 2021.

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