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Optimal consumption and investment for markets with random coefficients

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  • Belkacem Berdjane
  • Serguei Pergamenshchikov

Abstract

We consider an optimal investment and consumption problem for a Black–Scholes financial market with stochastic coefficients driven by a diffusion process. We assume that an agent makes consumption and investment decisions based on CRRA utility functions. The dynamic programming approach leads to an investigation of the Hamilton–Jacobi–Bellman (HJB) equation which is a highly nonlinear partial differential equation (PDE) of the second order. By using the Feynman–Kac representation, we prove uniqueness and smoothness of the solution. Moreover, we study the optimal convergence rate of iterative numerical schemes for both the value function and the optimal portfolio. We show that in this case, the optimal convergence rate is super-geometric, i.e., more rapid than any geometric one. We apply our results to a stochastic volatility financial market. Copyright Springer-Verlag 2013

Suggested Citation

  • Belkacem Berdjane & Serguei Pergamenshchikov, 2013. "Optimal consumption and investment for markets with random coefficients," Finance and Stochastics, Springer, vol. 17(2), pages 419-446, April.
    • Handle: RePEc:spr:finsto:v:17:y:2013:i:2:p:419-446
    DOI: 10.1007/s00780-012-0193-0
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    References listed on IDEAS

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    1. Thaleia Zariphopoulou, 2001. "A solution approach to valuation with unhedgeable risks," Finance and Stochastics, Springer, vol. 5(1), pages 61-82.
    2. Holger Kraft & Mogens Steffensen, 2006. "Portfolio problems stopping at first hitting time with application to default risk," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 63(1), pages 123-150, February.
    3. Hernández-Hernández Daniel & Schied Alexander, 2006. "Robust utility maximization in a stochastic factor model," Statistics & Risk Modeling, De Gruyter, vol. 24(1/2006), pages 1-17, July.
    4. {L}ukasz Delong & Claudia Kluppelberg, 2008. "Optimal investment and consumption in a Black--Scholes market with L\'evy-driven stochastic coefficients," Papers 0806.2570, arXiv.org.
    Full references (including those not matched with items on IDEAS)

    Citations

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    Cited by:

    1. Kraft, Holger & Seiferling, Thomas & Seifried, Frank Thomas, 2016. "Optimal consumption and investment with Epstein-Zin recursive utility," SAFE Working Paper Series 52, Leibniz Institute for Financial Research SAFE, revised 2016.
    2. Belkacem Berdjane & Sergei Pergamenshchikov, 2012. "Sequential $\delta$-optimal consumption and investment for stochastic volatility markets with unknown parameters," Papers 1210.5111, arXiv.org, revised May 2015.
    3. Evgeny Pchelintsev & Serguei Pergamenshchikov & Maria Leshchinskaya, 2022. "Improved estimation method for high dimension semimartingale regression models based on discrete data," Statistical Inference for Stochastic Processes, Springer, vol. 25(3), pages 537-576, October.
    4. Dariusz Zawisza, 2020. "On the parabolic equation for portfolio problems," Papers 2003.13317, arXiv.org, revised Oct 2020.
    5. Rodwell Kufakunesu & Calisto Guambe, 2018. "On the optimal investment-consumption and life insurance selection problem with an external stochastic factor," Papers 1808.04608, arXiv.org.
    6. Chen, Xu & Yang, Xiang-qun, 2015. "Optimal consumption and investment problem with random horizon in a BMAP model," Insurance: Mathematics and Economics, Elsevier, vol. 61(C), pages 197-205.
    7. Dariusz Zawisza, 2016. "Smooth solutions to discounted reward control problems with unbounded discount rate and financial applications," Papers 1602.00899, arXiv.org, revised Feb 2016.
    8. Sahar Albosaily & Serguei Pergamenchtchikov, 2021. "Optimal Investment and Consumption for Multidimensional Spread Financial Markets with Logarithmic Utility," Stats, MDPI, vol. 4(4), pages 1-15, November.
    9. Shuenn-Jyi Sheu & Li-Hsien Sun & Zheng Zhang, 2018. "Portfolio Optimization with Delay Factor Models," Papers 1805.01118, arXiv.org.
    10. 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.
    11. Sahar Albosaily & Serguei Pergamenshchikov, 2018. "Optimal investment and consumption for Ornstein-Uhlenbeck spread financial markets with logarithmic utility," Papers 1809.08139, arXiv.org.
    12. Yalc{c}in Aktar & Erik Taflin, 2014. "A remark on smooth solutions to a stochastic control problem with a power terminal cost function and stochastic volatilities," Papers 1405.3566, arXiv.org.

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    More about this item

    Keywords

    Black–Scholes market; Stochastic volatility; Optimal consumption and investment; Hamilton–Jacobi–Bellman equation; Feynman–Kac formula; Fixed-point solution; 91G10; 91G80; 93E20; G11;
    All these keywords.

    JEL classification:

    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions

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