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Capacitary measures for completely monotone kernels via singular control

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  • Aur'elien Alfonsi

    (CERMICS)

  • Alexander Schied

Abstract

We give a singular control approach to the problem of minimizing an energy functional for measures with given total mass on a compact real interval, when energy is defined in terms of a completely monotone kernel. This problem occurs both in potential theory and when looking for optimal financial order execution strategies under transient price impact. In our setup, measures or order execution strategies are interpreted as singular controls, and the capacitary measure is the unique optimal control. The minimal energy, or equivalently the capacity of the underlying interval, is characterized by means of a nonstandard infinite-dimensional Riccati differential equation, which is analyzed in some detail. We then show that the capacitary measure has two Dirac components at the endpoints of the interval and a continuous Lebesgue density in between. This density can be obtained as the solution of a certain Volterra integral equation of the second kind.

Suggested Citation

  • Aur'elien Alfonsi & Alexander Schied, 2012. "Capacitary measures for completely monotone kernels via singular control," Papers 1201.2756, arXiv.org, revised Feb 2013.
    • Handle: RePEc:arx:papers:1201.2756
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    References listed on IDEAS

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    1. Aur'elien Alfonsi & Antje Fruth & Alexander Schied, 2007. "Optimal execution strategies in limit order books with general shape functions," Papers 0708.1756, arXiv.org, revised Feb 2010.
    2. Aurelien Alfonsi & Antje Fruth & Alexander Schied, 2010. "Optimal execution strategies in limit order books with general shape functions," Quantitative Finance, Taylor & Francis Journals, vol. 10(2), pages 143-157.
    3. Obizhaeva, Anna A. & Wang, Jiang, 2013. "Optimal trading strategy and supply/demand dynamics," Journal of Financial Markets, Elsevier, vol. 16(1), pages 1-32.
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    Citations

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    Cited by:

    1. Iacopo Mastromatteo & Michael Benzaquen & Zoltan Eisler & Jean-Philippe Bouchaud, 2017. "Trading Lightly: Cross-Impact and Optimal Portfolio Execution," Papers 1702.03838, arXiv.org, revised Aug 2017.
    2. Gianbiagio Curato & Jim Gatheral & Fabrizio Lillo, 2014. "Optimal execution with nonlinear transient market impact," Papers 1412.4839, arXiv.org.
    3. Aurélien Alfonsi & Pierre Blanc, 2016. "Dynamic optimal execution in a mixed-market-impact Hawkes price model," Finance and Stochastics, Springer, vol. 20(1), pages 183-218, January.
    4. Eduardo Abi Jaber & Eyal Neuman, 2022. "Optimal Liquidation with Signals: the General Propagator Case," Working Papers hal-03835948, HAL.
    5. Eduardo Abi Jaber, 2020. "The Laplace transform of the integrated Volterra Wishart process," Working Papers hal-02367200, HAL.
    6. Eduardo Abi Jaber & Eyal Neuman, 2022. "Optimal Liquidation with Signals: the General Propagator Case," Papers 2211.00447, arXiv.org.
    7. Gianbiagio Curato & Jim Gatheral & Fabrizio Lillo, 2017. "Optimal execution with non-linear transient market impact," Quantitative Finance, Taylor & Francis Journals, vol. 17(1), pages 41-54, January.
    8. Aurélien Alfonsi & Pierre Blanc, 2016. "Dynamic optimal execution in a mixed-market-impact Hawkes price model," Finance and Stochastics, Springer, vol. 20(1), pages 183-218, January.
    9. Eduardo Abi Jaber, 2022. "The Laplace transform of the integrated Volterra Wishart process," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-02367200, HAL.
    10. Elias Strehle, 2016. "Optimal Execution in a Multiplayer Model of Transient Price Impact," Papers 1609.00599, arXiv.org, revised Mar 2019.
    11. Alexander Schied & Elias Strehle, 2017. "On the minimizers of energy forms with completely monotone kernel," Papers 1706.04844, arXiv.org, revised Aug 2018.
    12. Christopher Lorenz & Alexander Schied, 2013. "Drift dependence of optimal trade execution strategies under transient price impact," Finance and Stochastics, Springer, vol. 17(4), pages 743-770, October.
    13. Aur'elien Alfonsi & Pierre Blanc, 2014. "Dynamic optimal execution in a mixed-market-impact Hawkes price model," Papers 1404.0648, arXiv.org, revised Jun 2015.
    14. Eduardo Abi Jaber, 2022. "The Laplace transform of the integrated Volterra Wishart process," Post-Print hal-02367200, HAL.
    15. Aur'elien Alfonsi & Pierre Blanc, 2015. "Extension and calibration of a Hawkes-based optimal execution model," Papers 1506.08740, arXiv.org.
    16. Eduardo Abi Jaber, 2019. "The Laplace transform of the integrated Volterra Wishart process," Papers 1911.07719, arXiv.org, revised Jul 2024.
    17. Jonathan Donier & Julius Bonart, 2014. "A Million Metaorder Analysis of Market Impact on the Bitcoin," Papers 1412.4503, arXiv.org, revised Sep 2015.
    18. Alexander Schied & Elias Strehle & Tao Zhang, 2015. "High-frequency limit of Nash equilibria in a market impact game with transient price impact," Papers 1509.08281, arXiv.org, revised May 2017.
    19. Aurélien Alfonsi & Pierre Blanc, 2016. "Dynamic optimal execution in a mixed-market-impact Hawkes price model," Post-Print hal-00971369, HAL.
    20. Eduardo Abi Jaber, 2022. "The Laplace transform of the integrated Volterra Wishart process," Mathematical Finance, Wiley Blackwell, vol. 32(1), pages 309-348, January.
    21. Tao Chen & Mike Ludkovski & Moritz Vo{ss}, 2022. "On Parametric Optimal Execution and Machine Learning Surrogates," Papers 2204.08581, arXiv.org, revised Oct 2023.

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