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Homogenization and asymptotics for small transaction costs: the multidimensional case

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  • Dylan Possamai
  • H. Mete Soner
  • Nizar Touzi

Abstract

In the context of the multi-dimensional infinite horizon optimal consumption-investment problem with proportional transaction costs, we provide the first order expansion in small transact costs. Similar to the one-dimensional derivation in our accompanying paper [42], the asymptotic expansion is expressed in terms of a singular ergodic control problem, and our arguments are based on the theory of viscosity solutions, and the techniques of homogenization which leads to a system of corrector equations. In contrast with the one-dimensional case, no explicit solution of the first corrector equation is available anymore. Finally, we provide some numerical results which illustrate the structure of the first order optimal controls.

Suggested Citation

  • Dylan Possamai & H. Mete Soner & Nizar Touzi, 2012. "Homogenization and asymptotics for small transaction costs: the multidimensional case," Papers 1212.6275, arXiv.org, revised Jan 2013.
  • Handle: RePEc:arx:papers:1212.6275
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    References listed on IDEAS

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    1. George M. Constantinides, 2005. "Capital Market Equilibrium with Transaction Costs," World Scientific Book Chapters,in: Theory Of Valuation, chapter 7, pages 207-227 World Scientific Publishing Co. Pte. Ltd..
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    5. Stefan Gerhold & Johannes Muhle-Karbe & Walter Schachermayer, 2010. "Asymptotics and Duality for the Davis and Norman Problem," Papers 1010.0627, arXiv.org, revised Aug 2011.
    6. C. Atkinson & S. Mokkhavesa, 2004. "Multi-asset portfolio optimization with transaction cost," Applied Mathematical Finance, Taylor & Francis Journals, vol. 11(2), pages 95-123.
    7. Dumas, Bernard & Luciano, Elisa, 1991. " An Exact Solution to a Dynamic Portfolio Choice Problem under Transactions Costs," Journal of Finance, American Finance Association, vol. 46(2), pages 577-595, June.
    8. Kumar Muthuraman & Sunil Kumar, 2006. "Multidimensional Portfolio Optimization With Proportional Transaction Costs," Mathematical Finance, Wiley Blackwell, vol. 16(2), pages 301-335.
    9. H. Mete Soner & Nizar Touzi, 2012. "Homogenization and asymptotics for small transaction costs," Papers 1202.6131, arXiv.org, revised Jun 2013.
    10. Halil Mete Soner & Guy Barles, 1998. "Option pricing with transaction costs and a nonlinear Black-Scholes equation," Finance and Stochastics, Springer, vol. 2(4), pages 369-397.
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    Citations

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    Cited by:

    1. Filippo Passerini & Samuel E. Vazquez, 2015. "Optimal Trading with Alpha Predictors," Papers 1501.03756, arXiv.org, revised Jan 2015.
    2. Zura Kakushadze, 2015. "Combining Alphas via Bounded Regression," Risks, MDPI, Open Access Journal, vol. 3(4), pages 1-17, November.
    3. Albert Altarovici & Johannes Muhle-Karbe & Halil Soner, 2015. "Asymptotics for fixed transaction costs," Finance and Stochastics, Springer, vol. 19(2), pages 363-414, April.
    4. Ludovic Moreau & Johannes Muhle-Karbe & H. Mete Soner, 2014. "Trading with Small Price Impact," Papers 1402.5304, arXiv.org, revised Mar 2015.
    5. Bruno Bouchard & Johannes Muhle-Karbe, 2018. "Simple Bounds for Transaction Costs," Papers 1802.06120, arXiv.org.
    6. David Hobson & Alex S. L. Tse & Yeqi Zhu, 2016. "A multi-asset investment and consumption problem with transaction costs," Papers 1612.01327, arXiv.org.
    7. Johannes Muhle-Karbe & Max Reppen & H. Mete Soner, 2016. "A Primer on Portfolio Choice with Small Transaction Costs," Papers 1612.01302, arXiv.org, revised May 2017.
    8. Bruno Bouchard & Ludovic Moreau & Mete Soner, 2016. "Hedging under an expected loss constraint with small transaction costs," Post-Print hal-00863562, HAL.
    9. Kühn, Christoph & Muhle-Karbe, Johannes, 2015. "Optimal liquidity provision," Stochastic Processes and their Applications, Elsevier, vol. 125(7), pages 2493-2515.
    10. Ibrahim Ekren & Johannes Muhle-Karbe, 2017. "Portfolio Choice with Small Temporary and Transient Price Impact," Papers 1705.00672, arXiv.org, revised Mar 2018.
    11. Albert Altarovici & Max Reppen & H. Mete Soner, 2016. "Optimal Consumption and Investment with Fixed and Proportional Transaction Costs," Papers 1610.03958, arXiv.org.
    12. Jiatu Cai & Masaaki Fukasawa, 2016. "Asymptotic replication with modified volatility under small transaction costs," Finance and Stochastics, Springer, vol. 20(2), pages 381-431, April.
    13. Arash Fahim & Wan-Yu Tsai, 2017. "A Numerical Scheme for A Singular control problem: Investment-Consumption Under Proportional Transaction Costs," Papers 1711.01017, arXiv.org.
    14. Ren Liu & Johannes Muhle-Karbe & Marko H. Weber, 2014. "Rebalancing with Linear and Quadratic Costs," Papers 1402.5306, arXiv.org, revised Sep 2017.
    15. Ibrahim Ekren & Ren Liu & Johannes Muhle-Karbe, 2015. "Optimal Rebalancing Frequencies for Multidimensional Portfolios," Papers 1510.05097, arXiv.org, revised Sep 2017.
    16. Jiatu Cai & Mathieu Rosenbaum & Peter Tankov, 2015. "Asymptotic Lower Bounds for Optimal Tracking: a Linear Programming Approach," Papers 1510.04295, arXiv.org.
    17. Bruno Bouchard & Johannes Muhle-Karbe, 2018. "Simple Bounds for Transaction Costs," Working Papers hal-01711371, HAL.
    18. Jiatu Cai & Masaaki Fukasawa, 2014. "Asymptotic replication with modified volatility under small transaction costs," Papers 1408.5677, arXiv.org.
    19. Bruno Bouchard & Masaaki Fukasawa & Martin Herdegen & Johannes Muhle-Karbe, 2018. "Equilibrium Returns with Transaction Costs," Post-Print hal-01569408, HAL.
    20. Framstad, Nils Chr., 2014. "The Effect of Small Intervention Costs on the Optimal Extraction of Dividends and Renewable Resources in a Jump-Diffusion Model," Memorandum 25/2014, Oslo University, Department of Economics.

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