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

Author

Listed:
  • 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)

Abstract

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
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    Citations

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    Cited by:

    1. Jan Kallsen & Johannes Muhle-Karbe, 2009. "Utility maximization in models with conditionally independent increments," Papers 0911.3608, arXiv.org.
    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

    Keywords

    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

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