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On Feedback Control in Kelly Betting: An Approximation Approach

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  • Chung-Han Hsieh

Abstract

In this paper, we consider a simple discrete-time optimal betting problem using the celebrated Kelly criterion, which calls for maximization of the expected logarithmic growth of wealth. While the classical Kelly betting problem can be solved via standard concave programming technique, an alternative but attractive approach is to invoke a Taylor-based approximation, which recasts the problem into quadratic programming and obtain the closed-form approximate solution. The focal point of this paper is to fill some voids in the existing results by providing some interesting properties when such an approximate solution is used. Specifically, the best achievable betting performance, positivity of expected cumulative gain or loss and its associated variance, expected growth property, variance of logarithmic growth, and results related to the so-called survivability (no bankruptcy) are provided.

Suggested Citation

  • Chung-Han Hsieh, 2020. "On Feedback Control in Kelly Betting: An Approximation Approach," Papers 2004.14048, arXiv.org, revised May 2020.
    • Handle: RePEc:arx:papers:2004.14048
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    References listed on IDEAS

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    1. Leonard Maclean & Edward Thorp & William Ziemba, 2010. "Long-term capital growth: the good and bad properties of the Kelly and fractional Kelly capital growth criteria," Quantitative Finance, Taylor & Francis Journals, vol. 10(7), pages 681-687.
    2. Henry Allen Latane, 1959. "Criteria for Choice Among Risky Ventures," Journal of Political Economy, University of Chicago Press, vol. 67(2), pages 144-144.
    3. Chung-Han Hsieh & B. Ross Barmish & John A. Gubner, 2019. "On Positive Solutions of a Delay Equation Arising When Trading in Financial Markets," Papers 1901.02480, arXiv.org, revised Oct 2019.
    4. 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.
    5. Chung-Han Hsieh & B. Ross Barmish & John A. Gubner, 2018. "At What Frequency Should the Kelly Bettor Bet?," Papers 1801.06737, arXiv.org, revised Aug 2018.
    6. Chung-Han Hsieh & John A. Gubner & B. Ross Barmish, 2018. "Rebalancing Frequency Considerations for Kelly-Optimal Stock Portfolios in a Control-Theoretic Framework," Papers 1807.05265, arXiv.org, revised Aug 2018.
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    Cited by:

    1. Chung-Han Hsieh, 2022. "On Robust Optimal Linear Feedback Stock Trading," Papers 2202.02300, arXiv.org.
    2. Chung-Han Hsieh, 2022. "On Solving Robust Log-Optimal Portfolio: A Supporting Hyperplane Approximation Approach," Papers 2202.03858, arXiv.org.

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