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On Positive Solutions of a Delay Equation Arising When Trading in Financial Markets

Author

Listed:
  • Chung-Han Hsieh
  • B. Ross Barmish
  • John A. Gubner

Abstract

We consider a discrete-time, linear state equation with delay which arises as a model for a trader's account value when buying and selling a risky asset in a financial market. The state equation includes a nonnegative feedback gain $\alpha$ and a sequence $v(k)$ which models asset returns which are within known bounds but otherwise arbitrary. We introduce two thresholds, $\alpha_-$ and $\alpha_+$, depending on these bounds, and prove that for $\alpha \alpha_+$, we show that there is always a sequence of asset returns for which the state fails to be positive for all time; i.e., along this sequence, bankruptcy is certain and the solution of the state equation ceases to be meaningful after some finite time. Finally, this paper also includes a conjecture which says that for the "gap" interval $\alpha_- \leq \alpha \leq \alpha_+,$ state positivity is also guaranteed for all time. Support for the conjecture, both theoretical and computational, is provided.

Suggested Citation

  • 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.
    • Handle: RePEc:arx:papers:1901.02480
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    File URL: http://arxiv.org/pdf/1901.02480
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    References listed on IDEAS

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    1. 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.
    2. 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, 2020. "On Feedback Control in Kelly Betting: An Approximation Approach," Papers 2004.14048, arXiv.org, revised May 2020.
    2. Chung-Han Hsieh, 2022. "On Robustness of Double Linear Trading with Transaction Costs," Papers 2209.12383, arXiv.org.
    3. Chung-Han Hsieh, 2020. "Necessary and Sufficient Conditions for Frequency-Based Kelly Optimal Portfolio," Papers 2004.12099, arXiv.org.
    4. Chung-Han Hsieh, 2020. "Generalization of Affine Feedback Stock Trading Results to Include Stop-Loss Orders," Papers 2004.12848, arXiv.org.

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    1. Chung-Han Hsieh, 2020. "On Feedback Control in Kelly Betting: An Approximation Approach," Papers 2004.14048, arXiv.org, revised May 2020.
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