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Homogenization and asymptotics for small transaction costs: the multidimensional case

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  • Dylan Possama\"i
  • H. Mete Soner
  • Nizar Touzi
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    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.

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    File URL: http://arxiv.org/pdf/1212.6275
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    Paper provided by arXiv.org in its series Papers with number 1212.6275.

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    Date of creation: Dec 2012
    Date of revision: Jan 2013
    Handle: RePEc:arx:papers:1212.6275

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    1. Magill, Michael J. P. & Constantinides, George M., 1976. "Portfolio selection with transactions costs," Journal of Economic Theory, Elsevier, Elsevier, vol. 13(2), pages 245-263, October.
    2. Karel Janeček & Steven Shreve, 2004. "Asymptotic analysis for optimal investment and consumption with transaction costs," Finance and Stochastics, Springer, Springer, vol. 8(2), pages 181-206, 05.
    3. Kumar Muthuraman & Sunil Kumar, 2006. "Multidimensional Portfolio Optimization With Proportional Transaction Costs," Mathematical Finance, Wiley Blackwell, Wiley Blackwell, vol. 16(2), pages 301-335.
    4. C. Atkinson & S. Mokkhavesa, 2004. "Multi-asset portfolio optimization with transaction cost," Applied Mathematical Finance, Taylor & Francis Journals, Taylor & Francis Journals, vol. 11(2), pages 95-123.
    5. H. Mete Soner & Nizar Touzi, 2012. "Homogenization and asymptotics for small transaction costs," Papers 1202.6131, arXiv.org, revised Jun 2013.
    6. Halil Mete Soner & Guy Barles, 1998. "Option pricing with transaction costs and a nonlinear Black-Scholes equation," Finance and Stochastics, Springer, Springer, vol. 2(4), pages 369-397.
    7. 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.
    8. Dumas, Bernard & Luciano, Elisa, 1991. " An Exact Solution to a Dynamic Portfolio Choice Problem under Transactions Costs," Journal of Finance, American Finance Association, American Finance Association, vol. 46(2), pages 577-95, June.
    9. Constantinides, George M, 1986. "Capital Market Equilibrium with Transaction Costs," Journal of Political Economy, University of Chicago Press, University of Chicago Press, vol. 94(4), pages 842-62, August.
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    Cited by:
    1. Ludovic Moreau & Johannes Muhle-Karbe & H. Mete Soner, 2014. "Trading with Small Price Impact," Papers 1402.5304, arXiv.org.
    2. Jiatu Cai & Masaaki Fukasawa, 2014. "Asymptotic replication with modified volatility under small transaction costs," Papers 1408.5677, arXiv.org.
    3. Ren Liu & Johannes Muhle-Karbe & Marko Weber, 2014. "Rebalancing with Linear and Quadratic Costs," Papers 1402.5306, arXiv.org.

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