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Optimal portfolio management rules in a non-Gaussian market with durability and intertemporal substitution


  • Kristin Reikvam

    () (Department of Mathematics, University of Oslo, P.O. Box 1053, Blindern, N-0316 Oslo, Norway and MaPhySto - Centre for Mathematical Physics and Stochastics, University of Aarhus, Ny Munkegade, DK-8000 Å rhus, Denmark

  • Fred Espen Benth

    () (Department of Mathematics, University of Oslo, P.O. Box 1053, Blindern, N-0316 Oslo, Norway and MaPhySto - Centre for Mathematical Physics and Stochastics, University of Aarhus, Ny Munkegade, DK-8000 Å rhus, Denmark

  • Kenneth Hvistendahl Karlsen

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


We consider an optimal portfolio-consumption problem which incorporates the notions of durability and intertemporal substitution. The logreturns of the uncertain assets are not necessarily normally distributed. The natural models then involve Lévy processes as driving noise instead of the more frequently used Brownian motion. The optimization problem is a singular stochastic control problem and the associated Hamilton-Jacobi-Bellman equation is a nonlinear second order degenerate elliptic integro-differential equation subject to gradient and state constraints. For utility functions of HARA type, we calculate the optimal investment and consumption policies together with an explicit expression for the value function when the Lévy process has only negative jumps. For the classical Merton problem, which is a special case of our optimization problem, we provide explicit policies for general Lévy processes having both positive and negative jumps. Instead of following the classical approach of using a verification theorem, we validate our solution candidates within a viscosity solution framework. To this end, the value function of our singular control problem is characterized as the unique constrained viscosity solution of the Hamilton-Jacobi-Bellman equation in the case of general utilities and general Lévy processes.

Suggested Citation

  • 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.
  • Handle: RePEc:spr:finsto:v:5:y:2001:i:4:p:447-467
    Note: received: April 2000; final version received: July 2000

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    Cited by:

    1. Peter Bank & Helena Kauppila, 2014. "Convex duality for stochastic singular control problems," Papers 1407.7717,
    2. 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.
    3. Motoh, Tsujimura, 2004. "Optimal natural resources management under uncertainty with catastrophic risk," Energy Economics, Elsevier, vol. 26(3), pages 487-499, May.
    4. 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.
    5. Guambe, Calisto & Kufakunesu, Rodwell, 2015. "A note on optimal investment–consumption–insurance in a Lévy market," Insurance: Mathematics and Economics, Elsevier, vol. 65(C), pages 30-36.
    6. Seydel, Roland C., 2009. "Existence and uniqueness of viscosity solutions for QVI associated with impulse control of jump-diffusions," Stochastic Processes and their Applications, Elsevier, vol. 119(10), pages 3719-3748, October.

    More about this item


    Portfolio choice; intertemporal substitution; singular stochastic control; dynamic programming method; integro-differential variational inequality; viscosity solution; closed form solution;

    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


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