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Consumption-portfolio optimization with recursive utility in incomplete markets


  • Holger Kraft


  • Frank Seifried


  • Mogens Steffensen



In an incomplete market, we study the optimal consumption-portfolio decision of an investor with recursive preferences of Epstein–Zin type. Applying a classical dynamic programming approach, we formulate the associated Hamilton–Jacobi–Bellman equation and provide a suitable verification theorem. The proof of this verification theorem is complicated by the fact that the Epstein–Zin aggregator is non-Lipschitz, so standard verification results (e.g. in Duffie and Epstein, Econometrica 60, 393–394, 1992 ) are not applicable. We provide new explicit solutions to the Bellman equation with Epstein–Zin preferences in an incomplete market for non-unit elasticity of intertemporal substitution (EIS) and apply our verification result to prove that they solve the consumption-investment problem. We also compare our exact solutions to the Campbell–Shiller approximation and assess its accuracy. Copyright Springer-Verlag 2013

Suggested Citation

  • Holger Kraft & Frank Seifried & Mogens Steffensen, 2013. "Consumption-portfolio optimization with recursive utility in incomplete markets," Finance and Stochastics, Springer, vol. 17(1), pages 161-196, January.
  • Handle: RePEc:spr:finsto:v:17:y:2013:i:1:p:161-196
    DOI: 10.1007/s00780-012-0184-1

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    References listed on IDEAS

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    Cited by:

    1. Dietmar Leisen & Eckhard Platen, 2017. "Investing for the Long Run," Papers 1705.03929,
    2. Hao Xing, 2017. "Consumption–investment optimization with Epstein–Zin utility in incomplete markets," Finance and Stochastics, Springer, vol. 21(1), pages 227-262, January.
    3. Jensen, N.R. & Steffensen, M., 2015. "Personal finance and life insurance under separation of risk aversion and elasticity of substitution," Insurance: Mathematics and Economics, Elsevier, vol. 62(C), pages 28-41.
    4. Dietmar P.J. Leisen & Eckhard Platen, 2017. "Investing for the Long Run," Research Paper Series 381, Quantitative Finance Research Centre, University of Technology, Sydney.
    5. Anis Matoussi & Hao Xing, 2016. "Convex duality for stochastic differential utility," Papers 1601.03562,
    6. Hao Xing, 2015. "Consumption investment optimization with Epstein-Zin utility in incomplete markets," Papers 1501.04747,, revised Nov 2015.
    7. Holger Kraft & Thomas Seiferling & Frank Thomas Seifried, 2017. "Optimal consumption and investment with Epstein–Zin recursive utility," Finance and Stochastics, Springer, vol. 21(1), pages 187-226, January.

    More about this item


    Consumption-portfolio optimization; Recursive utility; Stochastic control approach; Stochastic volatility; Unspanned state process; Campbell–Shiller approximation; 93E20; 91G10; G11; D91; C61;

    JEL classification:

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


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