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An elementary approach to the Merton problem

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  • Martin Herdegen
  • David Hobson
  • Joseph Jerome

Abstract

In this article we consider the infinite‐horizon Merton investment‐consumption problem in a constant‐parameter Black–Scholes–Merton market for an agent with constant relative risk aversion R. The classical primal approach is to write down a candidate value function and to use a verification argument to prove that this is the solution to the problem. However, features of the problem take it outside the standard settings of stochastic control, and the existing primal verification proofs rely on parameter restrictions (especially, but not only, R

Suggested Citation

  • Martin Herdegen & David Hobson & Joseph Jerome, 2021. "An elementary approach to the Merton problem," Mathematical Finance, Wiley Blackwell, vol. 31(4), pages 1218-1239, October.
    • Handle: RePEc:bla:mathfi:v:31:y:2021:i:4:p:1218-1239
    DOI: 10.1111/mafi.12311
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    References listed on IDEAS

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    1. Merton, Robert C., 1971. "Optimum consumption and portfolio rules in a continuous-time model," Journal of Economic Theory, Elsevier, vol. 3(4), pages 373-413, December.
    2. Ioannis Karatzas & John P. Lehoczky & Suresh P. Sethi & Steven E. Shreve, 1986. "Explicit Solution of a General Consumption/Investment Problem," Mathematics of Operations Research, INFORMS, vol. 11(2), pages 261-294, May.
    3. M. H. A. Davis & A. R. Norman, 1990. "Portfolio Selection with Transaction Costs," Mathematics of Operations Research, INFORMS, vol. 15(4), pages 676-713, November.
    4. Martin Herdegen, 2017. "No-Arbitrage In A Numéraire-Independent Modeling Framework," Mathematical Finance, Wiley Blackwell, vol. 27(2), pages 568-603, April.
    5. J. Lehoczky & S. Sethi & S. Shreve, 1983. "Optimal Consumption and Investment Policies Allowing Consumption Constraints and Bankruptcy," Mathematics of Operations Research, INFORMS, vol. 8(4), pages 613-636, November.
    6. Merton, Robert C, 1969. "Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case," The Review of Economics and Statistics, MIT Press, vol. 51(3), pages 247-257, August.
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    Cited by:

    1. Martin Herdegen & David Hobson & Joseph Jerome, 2023. "The infinite-horizon investment–consumption problem for Epstein–Zin stochastic differential utility. II: Existence, uniqueness and verification for ϑ ∈ ( 0 , 1 ) $\vartheta \in (0,1)$," Finance and Stochastics, Springer, vol. 27(1), pages 159-188, January.
    2. Alain Bensoussan & Ka Chun Cheung & Yiqun Li & Sheung Chi Phillip Yam, 2022. "Inter‐temporal mutual‐fund management," Mathematical Finance, Wiley Blackwell, vol. 32(3), pages 825-877, July.
    3. 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.
    4. Martin Herdegen & David Hobson & Alex S. L. Tse, 2024. "Portfolio Optimization under Transaction Costs with Recursive Preferences," Papers 2402.08387, arXiv.org.
    5. Martin Herdegen & David Hobson & Joseph Jerome, 2021. "Proper solutions for Epstein-Zin Stochastic Differential Utility," Papers 2112.06708, arXiv.org.

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