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Finding the nonnegative minimal solutions of Cauchy PDEs in a volatility-stabilized market

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  • Nicole Tianjiao Yang
  • Tomoyuki Ichiba

Abstract

The strong relative arbitrage problem in Stochastic Portfolio Theory seeks an investment strategy that almost surely outperforms a benchmark portfolio at the end of a given time horizon. The highest relative return in relative arbitrage opportunities is characterized by the smallest nonnegative continuous solution of a Cauchy problem for a partial differential equation (PDE). However, solving this type of PDE poses analytical and numerical challenges, due to the high dimensionality and its non-unique solutions. In this paper, we discuss numerical methods to address the relative arbitrage problem and the associated PDE in a volatility-stabilized market, using time-changed Bessel bridges. We present a practical algorithm and demonstrate numerical results through an example in volatility-stabilized markets.

Suggested Citation

  • Nicole Tianjiao Yang & Tomoyuki Ichiba, 2024. "Finding the nonnegative minimal solutions of Cauchy PDEs in a volatility-stabilized market," Papers 2411.13558, arXiv.org, revised May 2025.
    • Handle: RePEc:arx:papers:2411.13558
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    References listed on IDEAS

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    1. Daniel Fernholz & Ioannis Karatzas, 2010. "On optimal arbitrage," Papers 1010.4987, arXiv.org.
    2. Steven L. Heston & Mark Loewenstein & Gregory A. Willard, 2007. "Options and Bubbles," The Review of Financial Studies, Society for Financial Studies, vol. 20(2), pages 359-390.
    3. Nicole Tianjiao Yang & Tomoyuki Ichiba, 2023. "Relative Arbitrage Opportunities in an Extended Mean Field System," Papers 2311.02690, arXiv.org, revised Nov 2025.
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