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Turnpike property and convergence rate for an investment model with general utility functions

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Listed:
  • Bian, Baojun
  • Zheng, Harry

Abstract

In this paper we aim to address two questions faced by a long-term investor with a power-type utility at high levels of wealth: one is whether the turnpike property still holds for a general utility that is not necessarily differentiable or strictly concave, the other is whether the error and the convergence rate of the turnpike property can be estimated. We give positive answers to both questions. To achieve these results, we first show that there is a classical solution to the HJB equation and give a representation of the solution in terms of the dual function of the solution to the dual HJB equation. We demonstrate the usefulness of that representation with some nontrivial examples that would be difficult to solve with the trial and error method. We then combine the dual method and the partial differential equation method to give a direct proof to the turnpike property and to estimate the error and the convergence rate of the optimal policy when the utility function is continuously differentiable and strictly concave. We finally relax the conditions of the utility function and provide some sufficient conditions that guarantee the turnpike property and the convergence rate in terms of both primal and dual utility functions.

Suggested Citation

  • Bian, Baojun & Zheng, Harry, 2015. "Turnpike property and convergence rate for an investment model with general utility functions," Journal of Economic Dynamics and Control, Elsevier, vol. 51(C), pages 28-49.
  • Handle: RePEc:eee:dyncon:v:51:y:2015:i:c:p:28-49
    DOI: 10.1016/j.jedc.2014.09.025
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    References listed on IDEAS

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    1. Dybvig, Philip H & Rogers, L C G & Back, Kerry, 1999. "Portfolio Turnpikes," The Review of Financial Studies, Society for Financial Studies, vol. 12(1), pages 165-195.
    2. Cox, John C. & Huang, Chi-fu, 1992. "A continuous-time portfolio turnpike theorem," Journal of Economic Dynamics and Control, Elsevier, vol. 16(3-4), pages 491-507.
    3. Jin, Xing, 1998. "Consumption and portfolio turnpike theorems in a continuous-time finance model1," Journal of Economic Dynamics and Control, Elsevier, vol. 22(7), pages 1001-1026, May.
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    Citations

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    Cited by:

    1. Dong, Yinghui & Zheng, Harry, 2019. "Optimal investment of DC pension plan under short-selling constraints and portfolio insurance," Insurance: Mathematics and Economics, Elsevier, vol. 85(C), pages 47-59.
    2. Tianran Geng & Thaleia Zariphopoulou, 2017. "Temporal and Spatial Turnpike-Type Results Under Forward Time-Monotone Performance Criteria," Papers 1702.05649, arXiv.org.
    3. Jingtang Ma & Wenyuan Li & Harry Zheng, 2017. "Dual control Monte Carlo method for tight bounds of value function under Heston stochastic volatility model," Papers 1710.10487, arXiv.org.
    4. Sigrid Kallblad & Thaleia Zariphopoulou, 2017. "On the Black's equation for the risk tolerance function," Papers 1705.07472, arXiv.org.
    5. Yusong Li & Harry Zheng, 2016. "Dynamic Convex Duality in Constrained Utility Maximization," Papers 1612.04407, arXiv.org.
    6. Ma, Jingtang & Li, Wenyuan & Zheng, Harry, 2017. "Dual control Monte-Carlo method for tight bounds of value function in regime switching utility maximization," European Journal of Operational Research, Elsevier, vol. 262(3), pages 851-862.
    7. Ashley Davey & Michael Monoyios & Harry Zheng, 2020. "Duality for optimal consumption with randomly terminating income," Papers 2011.00732, arXiv.org, revised May 2021.
    8. He, Yong & Zhou, Xia & Chen, Peimin & Wang, Xiaoyang, 2022. "An analytical solution for the robust investment-reinsurance strategy with general utilities," The North American Journal of Economics and Finance, Elsevier, vol. 63(C).
    9. Baojun Bian & Harry Zheng, 2018. "Turnpike Property and Convergence Rate for an Investment and Consumption Model," Papers 1808.04265, arXiv.org.
    10. Jingtang Ma & Jie Xing & Harry Zheng, 2018. "Global Closed-form Approximation of Free Boundary for Optimal Investment Stopping Problems," Papers 1810.09397, arXiv.org.

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    More about this item

    Keywords

    Non-strictly concave utility function; Smooth solution to HJB equation; Dual representation; Turnpike property; Convergence rate;
    All these keywords.

    JEL classification:

    • D9 - Microeconomics - - Micro-Based Behavioral Economics
    • G1 - Financial Economics - - General Financial Markets

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