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On A Class Of Rank-Based Continuous Semimartingales

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  • David Itkin
  • Martin Larsson

Abstract

Using the theory of Dirichlet forms we construct a large class of continuous semimartingales on an open domain $E \subset \mathbb{R}^d$, which are governed by rank-based, in addition to name-based, characteristics. Using the results of Baur et al. [Potential Analysis, 38(4):1233-1258,2013] we obtain a strong Feller property for this class of diffusions. As a consequence we are able to establish the nonexistence of triple collisions and obtain a simplified formula for the dynamics of its rank process. We also establish conditions under which the process is ergodic. Our main motivation is Stochastic Portfolio Theory (SPT), where rank-based diffusions of this type are used to model financial markets. We show that three main classes of models studied in SPT -- Atlas models, generalized volatility-stabilized models and polynomial models -- are special cases of our framework.

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  • David Itkin & Martin Larsson, 2021. "On A Class Of Rank-Based Continuous Semimartingales," Papers 2104.04396, arXiv.org.
    • Handle: RePEc:arx:papers:2104.04396
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    References listed on IDEAS

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    1. Shkolnikov, Mykhaylo, 2012. "Large systems of diffusions interacting through their ranks," Stochastic Processes and their Applications, Elsevier, vol. 122(4), pages 1730-1747.
    2. 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.
    3. David Itkin & Martin Larsson, 2020. "Robust Asymptotic Growth in Stochastic Portfolio Theory under Long-Only Constraints," Papers 2009.08533, arXiv.org, revised Aug 2021.
    4. Fernholz, Robert, 1999. "On the diversity of equity markets," Journal of Mathematical Economics, Elsevier, vol. 31(3), pages 393-417, April.
    5. Constantinos Kardaras & Scott Robertson, 2018. "Ergodic robust maximization of asymptotic growth," Papers 1801.06425, arXiv.org.
    6. Cuchiero, Christa, 2019. "Polynomial processes in stochastic portfolio theory," Stochastic Processes and their Applications, Elsevier, vol. 129(5), pages 1829-1872.
    7. Tomoyuki Ichiba & Vassilios Papathanakos & Adrian Banner & Ioannis Karatzas & Robert Fernholz, 2009. "Hybrid Atlas models," Papers 0909.0065, arXiv.org, revised Apr 2011.
    8. Radka Picková, 2014. "Generalized volatility-stabilized processes," Annals of Finance, Springer, vol. 10(1), pages 101-125, February.
    9. Damir Filipović & Martin Larsson, 2016. "Polynomial diffusions and applications in finance," Finance and Stochastics, Springer, vol. 20(4), pages 931-972, October.
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