IDEAS home Printed from
   My bibliography  Save this article

Optimal portfolio selection with consumption and nonlinear integro-differential equations with gradient constraint: A viscosity solution approach


  • Kristin Reikvam

    (Department of Mathematics, University of Oslo, P. O. Box 1053, Blindern, N-0316 Oslo, Norway Manuscript)

  • Fred Espen Benth

    (Department of Mathematics, University of Oslo, P. O. Box 1053, Blindern, N-0316 Oslo, Norway)

  • Kenneth Hvistendahl Karlsen

    (Department of Mathematics, University of Bergen, Johs. Brunsgt. 12, N-5008 Bergen, Norway)


We study a problem of optimal consumption and portfolio selection in a market where the logreturns of the uncertain assets are not necessarily normally distributed. The natural models then involve pure-jump Lévy processes as driving noise instead of Brownian motion like in the Black and Scholes model. The state constrained optimization problem involves the notion of local substitution and is of singular type. The associated Hamilton-Jacobi-Bellman equation is a nonlinear first order integro-differential equation subject to gradient and state constraints. We characterize the value function of the singular stochastic control problem as the unique constrained viscosity solution of the associated Hamilton-Jacobi-Bellman equation. This characterization is obtained in two main steps. First, we prove that the value function is a constrained viscosity solution of an integro-differential variational inequality. Second, to ensure that the characterization of the value function is unique, we prove a new comparison (uniqueness) result for the state constraint problem for a class of integro-differential variational inequalities. In the case of HARA utility, it is possible to determine an explicit solution of our portfolio-consumption problem when the Lévy process posseses only negative jumps. This is, however, the topic of a companion paper [7].

Suggested Citation

  • Kristin Reikvam & Fred Espen Benth & Kenneth Hvistendahl Karlsen, 2001. "Optimal portfolio selection with consumption and nonlinear integro-differential equations with gradient constraint: A viscosity solution approach," Finance and Stochastics, Springer, vol. 5(3), pages 275-303.
  • Handle: RePEc:spr:finsto:v:5:y:2001:i:3:p:275-303
    Note: received: July 1999; final version received: April 2000

    Download full text from publisher

    File URL:
    Download Restriction: Access to the full text of the articles in this series is restricted

    As the access to this document is restricted, you may want to search for a different version of it.


    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.

    Cited by:

    1. Jan Kallsen & Johannes Muhle-Karbe, 2009. "Utility maximization in models with conditionally independent increments," Papers 0911.3608,
    2. Barbachan, José Santiago Fajardo, 2003. "Optimal Consumption and Investment with Lévy Processes," Revista Brasileira de Economia - RBE, FGV/EPGE - Escola Brasileira de Economia e Finanças, Getulio Vargas Foundation (Brazil), vol. 57(4), October.
    3. Johannes Temme, 2012. "Power utility maximization in exponential Lévy models: convergence of discrete-time to continuous-time maximizers," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 76(1), pages 21-41, August.
    4. Albrecher, Hansjörg & Thonhauser, Stefan, 2008. "Optimal dividend strategies for a risk process under force of interest," Insurance: Mathematics and Economics, Elsevier, vol. 43(1), pages 134-149, August.
    5. repec:spr:compst:v:76:y:2012:i:1:p:21-41 is not listed on IDEAS
    6. Azcue, Pablo & Muler, Nora, 2012. "Optimal dividend policies for compound Poisson processes: The case of bounded dividend rates," Insurance: Mathematics and Economics, Elsevier, vol. 51(1), pages 26-42.
    7. Zhu, Jinxia & Chen, Feng, 2015. "Dividend optimization under reserve constraints for the Cramér–Lundberg model compounded by force of interest," Economic Modelling, Elsevier, vol. 46(C), pages 142-156.
    8. Le Courtois, Olivier & Menoncin, Francesco, 2015. "Portfolio optimisation with jumps: Illustration with a pension accumulation scheme," Journal of Banking & Finance, Elsevier, vol. 60(C), pages 127-137.
    9. Shah, Sudhir A., 2005. "Optimal management of durable pollution," Journal of Economic Dynamics and Control, Elsevier, vol. 29(6), pages 1121-1164, June.

    More about this item


    Portfolio choice; intertemporal utility; stochastic control; singular control; dynamic programming; integro-differential variational inequality; state constraint problem; viscosity solution; comparison result;

    JEL classification:

    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions
    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • D91 - Microeconomics - - Micro-Based Behavioral Economics - - - Role and Effects of Psychological, Emotional, Social, and Cognitive Factors on Decision Making


    Access and download statistics


    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:finsto:v:5:y:2001:i:3:p:275-303. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Sonal Shukla) or (Rebekah McClure). General contact details of provider: .

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.