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Consumption-investment problem with transaction costs for Lévy-driven price processes


  • Dimitri Vallière

    (Université de Franche-Comté)

  • Yuri Kabanov

    (Université de Franche-Comté
    National Research University Higher School of Economics)

  • Emmanuel Lépinette

    (National Research University Higher School of Economics
    Université Paris Dauphine)


We consider an optimal control problem for a linear stochastic integro-differential equation with conic constraints on the phase variable and with the control of singular–regular type. Our setting includes consumption-investment problems for models of financial markets in the presence of proportional transaction costs, where the prices of the assets are given by a geometric Lévy process, and the investor is allowed to take short positions. We prove that the Bellman function of the problem is a viscosity solution of an HJB equation. A uniqueness theorem for the solution of the latter is established. Special attention is paid to the dynamic programming principle.

Suggested Citation

  • Dimitri Vallière & Yuri Kabanov & Emmanuel Lépinette, 2016. "Consumption-investment problem with transaction costs for Lévy-driven price processes," Finance and Stochastics, Springer, vol. 20(3), pages 705-740, July.
  • Handle: RePEc:spr:finsto:v:20:y:2016:i:3:d:10.1007_s00780-016-0303-5
    DOI: 10.1007/s00780-016-0303-5

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    References listed on IDEAS

    1. Yuri Kabanov, 2009. "Markets with Transaction Costs. Mathematical Theory," Post-Print hal-00488168, HAL.
    2. Merton, Robert C., 1971. "Optimum consumption and portfolio rules in a continuous-time model," Journal of Economic Theory, Elsevier, vol. 3(4), pages 373-413, December.
    3. Framstad, Nils Chr. & Oksendal, Bernt & Sulem, Agnes, 2001. "Optimal consumption and portfolio in a jump diffusion market with proportional transaction costs," Journal of Mathematical Economics, Elsevier, vol. 35(2), pages 233-257, April.
    4. M. H. A. Davis & A. R. Norman, 1990. "Portfolio Selection with Transaction Costs," Mathematics of Operations Research, INFORMS, vol. 15(4), pages 676-713, November.
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    6. Kristin Reikvam & Fred Espen Benth & Kenneth Hvistendahl Karlsen, 2001. "Optimal portfolio management rules in a non-Gaussian market with durability and intertemporal substitution," Finance and Stochastics, Springer, vol. 5(4), pages 447-467.
    7. Yuri Kabanov & Claudia Klüppelberg, 2004. "A geometric approach to portfolio optimization in models with transaction costs," Finance and Stochastics, Springer, vol. 8(2), pages 207-227, May.
    8. Jakša Cvitanić & Vassilis Polimenis & Fernando Zapatero, 2008. "Optimal portfolio allocation with higher moments," Annals of Finance, Springer, vol. 4(1), pages 1-28, January.
    9. Luciano Campi & Mark Owen, 2011. "Multivariate utility maximization with proportional transaction costs," Finance and Stochastics, Springer, vol. 15(3), pages 461-499, September.
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    11. Susanne Emmer & Claudia Klüppelberg, 2004. "Optimal portfolios when stock prices follow an exponential Lévy process," Finance and Stochastics, Springer, vol. 8(1), pages 17-44, January.
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    Cited by:

    1. Len Patrick Dominic M. Garces & Gerald H. L. Cheang, 2021. "A numerical approach to pricing exchange options under stochastic volatility and jump-diffusion dynamics," Quantitative Finance, Taylor & Francis Journals, vol. 21(12), pages 2025-2054, December.

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    More about this item


    Consumption-investment problem; Lévy process; Transaction costs; Bellman function; Dynamic programming; HJB equation; Lyapunov function;
    All these keywords.

    JEL classification:

    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing


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