IDEAS home Printed from https://ideas.repec.org/p/arx/papers/2107.06593.html
   My bibliography  Save this paper

The Infinite Horizon Investment-Consumption Problem for Epstein-Zin Stochastic Differential Utility

Author

Listed:
  • David Hobson
  • Martin Herdegen
  • Joseph Jerome

Abstract

In this article we consider the optimal investment-consumption problem for an agent with preferences governed by Epstein-Zin stochastic differential utility who invests in a constant-parameter Black-Scholes-Merton market. The paper has three main goals: first, to provide a detailed introduction to infinite-horizon Epstein-Zin stochastic differential utility, including a discussion of which parameter combinations lead to a well-formulated problem; second, to prove existence and uniqueness of infinite horizon Epstein-Zin stochastic differential utility under a restriction on the parameters governing the agent's risk aversion and temporal variance aversion; and third, to provide a verification argument for the candidate optimal solution to the investment-consumption problem among all admissible consumption streams. To achieve these goals, we introduce a slightly different formulation of Epstein-Zin stochastic differential utility to that which is traditionally used in the literature. This formulation highlights the necessity and appropriateness of certain restrictions on the parameters governing the stochastic differential utility function.

Suggested Citation

  • David Hobson & Martin Herdegen & Joseph Jerome, 2021. "The Infinite Horizon Investment-Consumption Problem for Epstein-Zin Stochastic Differential Utility," Papers 2107.06593, arXiv.org.
    • Handle: RePEc:arx:papers:2107.06593
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/2107.06593
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Mehra, Rajnish & Prescott, Edward C., 1985. "The equity premium: A puzzle," Journal of Monetary Economics, Elsevier, vol. 15(2), pages 145-161, March.
    2. George Chacko & Luis M. Viceira, 2005. "Dynamic Consumption and Portfolio Choice with Stochastic Volatility in Incomplete Markets," The Review of Financial Studies, Society for Financial Studies, vol. 18(4), pages 1369-1402.
    3. Larry G. Epstein & Stanley E. Zin, 2013. "Substitution, risk aversion and the temporal behavior of consumption and asset returns: A theoretical framework," World Scientific Book Chapters, in: Leonard C MacLean & William T Ziemba (ed.), HANDBOOK OF THE FUNDAMENTALS OF FINANCIAL DECISION MAKING Part I, chapter 12, pages 207-239, World Scientific Publishing Co. Pte. Ltd..
    4. Yaroslav Melnyk & Johannes Muhle‐Karbe & Frank Thomas Seifried, 2020. "Lifetime investment and consumption with recursive preferences and small transaction costs," Mathematical Finance, Wiley Blackwell, vol. 30(3), pages 1135-1167, July.
    5. Schroder, Mark & Skiadas, Costis, 1999. "Optimal Consumption and Portfolio Selection with Stochastic Differential Utility," Journal of Economic Theory, Elsevier, vol. 89(1), pages 68-126, November.
    6. Geoffard, Pierre-Yves, 1996. "Discounting and Optimizing: Capital Accumulation Problems as Variational Minmax Problems," Journal of Economic Theory, Elsevier, vol. 69(1), pages 53-70, April.
    7. Herdegen, Martin & Muhle-Karbe, Johannes, 2019. "Sensitivity of optimal consumption streams," Stochastic Processes and their Applications, Elsevier, vol. 129(6), pages 1964-1992.
    8. Matoussi, Anis & Xing, Hao, 2018. "Convex duality for Epstein-Zin stochastic differential utility," LSE Research Online Documents on Economics 82519, London School of Economics and Political Science, LSE Library.
    9. Duffie, Darrel & Lions, Pierre-Louis, 1992. "PDE solutions of stochastic differential utility," Journal of Mathematical Economics, Elsevier, vol. 21(6), pages 577-606.
    10. Anis Matoussi & Hao Xing, 2018. "Convex duality for Epstein–Zin stochastic differential utility," Mathematical Finance, Wiley Blackwell, vol. 28(4), pages 991-1019, October.
    11. Kraft, Holger & Seifried, Frank Thomas, 2014. "Stochastic differential utility as the continuous-time limit of recursive utility," Journal of Economic Theory, Elsevier, vol. 151(C), pages 528-550.
    12. 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.
    13. Duffie, Darrell & Epstein, Larry G, 1992. "Stochastic Differential Utility," Econometrica, Econometric Society, vol. 60(2), pages 353-394, March.
    14. Duffie, Darrell & Epstein, Larry G, 1992. "Asset Pricing with Stochastic Differential Utility," The Review of Financial Studies, Society for Financial Studies, vol. 5(3), pages 411-436.
    Full references (including those not matched with items on IDEAS)

    Citations

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


    Cited by:

    1. Martin Herdegen & David Hobson & Joseph Jerome, 2021. "Proper solutions for Epstein-Zin Stochastic Differential Utility," Papers 2112.06708, arXiv.org.
    2. Michael Monoyios & Oleksii Mostovyi, 2022. "Stability of the Epstein-Zin problem," Papers 2208.09895, arXiv.org, revised Apr 2023.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Zixin Feng & Dejian Tian, 2021. "Optimal consumption and portfolio selection with Epstein-Zin utility under general constraints," Papers 2111.09032, arXiv.org, revised May 2023.
    2. Martin Herdegen & David Hobson & Joseph Jerome, 2023. "The infinite-horizon investment–consumption problem for Epstein–Zin stochastic differential utility. I: Foundations," Finance and Stochastics, Springer, vol. 27(1), pages 127-158, January.
    3. Martin Herdegen & David Hobson & Joseph Jerome, 2021. "Proper solutions for Epstein-Zin Stochastic Differential Utility," Papers 2112.06708, arXiv.org.
    4. Dirk Becherer & Wilfried Kuissi-Kamdem & Olivier Menoukeu-Pamen, 2023. "Optimal consumption with labor income and borrowing constraints for recursive preferences," Working Papers hal-04017143, HAL.
    5. Shigeta, Yuki, 2020. "Gain/loss asymmetric stochastic differential utility," Journal of Economic Dynamics and Control, Elsevier, vol. 118(C).
    6. Li, Hanwu & Riedel, Frank & Yang, Shuzhen, 2024. "Optimal consumption for recursive preferences with local substitution — the case of certainty," Journal of Mathematical Economics, Elsevier, vol. 110(C).
    7. Joshua Aurand & Yu-Jui Huang, 2019. "Epstein-Zin Utility Maximization on a Random Horizon," Papers 1903.08782, arXiv.org, revised May 2023.
    8. Yaroslav Melnyk & Johannes Muhle‐Karbe & Frank Thomas Seifried, 2020. "Lifetime investment and consumption with recursive preferences and small transaction costs," Mathematical Finance, Wiley Blackwell, vol. 30(3), pages 1135-1167, July.
    9. Fahrenwaldt, Matthias Albrecht & Jensen, Ninna Reitzel & Steffensen, Mogens, 2020. "Nonrecursive separation of risk and time preferences," Journal of Mathematical Economics, Elsevier, vol. 90(C), pages 95-108.
    10. Chen, Xingjiang & Ruan, Xinfeng & Zhang, Wenjun, 2021. "Dynamic portfolio choice and information trading with recursive utility," Economic Modelling, Elsevier, vol. 98(C), pages 154-167.
    11. Dariusz Zawisza, 2020. "On the parabolic equation for portfolio problems," Papers 2003.13317, arXiv.org, revised Oct 2020.
    12. Benzoni, Luca & Collin-Dufresne, Pierre & Goldstein, Robert S., 2011. "Explaining asset pricing puzzles associated with the 1987 market crash," Journal of Financial Economics, Elsevier, vol. 101(3), pages 552-573, September.
    13. Anis Matoussi & Hao Xing, 2016. "Convex duality for stochastic differential utility," Papers 1601.03562, arXiv.org.
    14. Kraft, Holger & Munk, Claus & Weiss, Farina, 2022. "Bequest motives in consumption-portfolio decisions with recursive utility," Journal of Banking & Finance, Elsevier, vol. 138(C).
    15. Roche, Hervé, 2011. "Asset prices in an exchange economy when agents have heterogeneous homothetic recursive preferences and no risk free bond is available," Journal of Economic Dynamics and Control, Elsevier, vol. 35(1), pages 80-96, January.
    16. Shigeta, Yuki, 2022. "Quasi-hyperbolic discounting under recursive utility and consumption–investment decisions," Journal of Economic Theory, Elsevier, vol. 204(C).
    17. Kexin Chen & Kyunghyun Park & Hoi Ying Wong, 2024. "Robust dividend policy: Equivalence of Epstein-Zin and Maenhout preferences," Papers 2406.12305, arXiv.org.
    18. Zhao, Hui & Wang, Suxin, 2022. "Optimal investment and benefit adjustment problem for a target benefit pension plan with Cobb-Douglas utility and Epstein-Zin recursive utility," European Journal of Operational Research, Elsevier, vol. 301(3), pages 1166-1180.
    19. Michael Monoyios & Oleksii Mostovyi, 2022. "Stability of the Epstein-Zin problem," Papers 2208.09895, arXiv.org, revised Apr 2023.
    20. Dietmar Leisen & Eckhard Platen, 2017. "Investing for the Long Run," Papers 1705.03929, arXiv.org.

    More about this item

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    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:arx:papers:2107.06593. See general information about how to correct material in RePEc.

    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.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with 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.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: arXiv administrators (email available below). General contact details of provider: http://arxiv.org/ .

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

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.