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Ergodic robust maximization of asymptotic growth

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  • Constantinos Kardaras
  • Scott Robertson

Abstract

We consider the problem of robustly maximizing the growth rate of investor wealth in the presence of model uncertainty. Possible models are all those under which the assets' region $E$ and instantaneous covariation $c$ are known, and where additionally the assets are stable in that their occupancy time measures converge to a law with density $p$. This latter assumption is motivated by the observed stability of ranked relative market capitalizations for equity markets. We seek to identify the robust optimal growth rate, as well as a trading strategy which achieves this rate in all models. Under minimal assumptions upon $(E,c,p)$, we identify the robust growth rate with the Donsker-Varadhan rate function from occupancy time Large Deviations theory. We also prove existence of, and explicitly identify, the optimal trading strategy. We then apply our results in the case of drift uncertainty for ranked relative market capitalizations. Assuming regularity under symmetrization for the covariance and limiting density of the ranked capitalizations, we explicitly identify the robust optimal trading strategy in this setting.

Suggested Citation

  • Constantinos Kardaras & Scott Robertson, 2018. "Ergodic robust maximization of asymptotic growth," Papers 1801.06425, arXiv.org.
  • Handle: RePEc:arx:papers:1801.06425
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    References listed on IDEAS

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    1. Daniel Fernholz & Ioannis Karatzas, 2010. "On optimal arbitrage," Papers 1010.4987, arXiv.org.
    2. Constantinos Kardaras & Scott Robertson, 2010. "Robust maximization of asymptotic growth," Papers 1005.3454, arXiv.org, revised Aug 2012.
    3. Banner, Adrian D. & Ghomrasni, Raouf, 2008. "Local times of ranked continuous semimartingales," Stochastic Processes and their Applications, Elsevier, vol. 118(7), pages 1244-1253, July.
    4. Robert Fernholz, 2001. "Equity portfolios generated by functions of ranked market weights," Finance and Stochastics, Springer, vol. 5(4), pages 469-486.
    5. Kardaras, Constantinos & Robertson, Scott, 2012. "Robust maximization of asymptotic growth," LSE Research Online Documents on Economics 44994, London School of Economics and Political Science, LSE Library.
    6. Paolo Guasoni & Scott Robertson, 2012. "Portfolios and risk premia for the long run," Papers 1203.1399, arXiv.org.
    7. Erhan Bayraktar & Yu-Jui Huang, 2011. "Robust maximization of asymptotic growth under covariance uncertainty," Papers 1107.2988, arXiv.org, revised Sep 2013.
    8. Christa Cuchiero & Walter Schachermayer & Ting-Kam Leonard Wong, 2016. "Cover's universal portfolio, stochastic portfolio theory and the numeraire portfolio," Papers 1611.09631, arXiv.org.
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    Cited by:

    1. David Itkin & Martin Larsson, 2020. "Robust Asymptotic Growth in Stochastic Portfolio Theory under Long-Only Constraints," Papers 2009.08533, arXiv.org, revised Aug 2021.
    2. David Itkin & Martin Larsson, 2021. "On A Class Of Rank-Based Continuous Semimartingales," Papers 2104.04396, arXiv.org.
    3. Andrew L. Allan & Christa Cuchiero & Chong Liu & David J. Promel, 2021. "Model-free Portfolio Theory: A Rough Path Approach," Papers 2109.01843, arXiv.org, revised Oct 2022.

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