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

Nicole Tianjiao Yang and Tomoyuki Ichiba

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Abstract: The strong relative arbitrage problem in Stochastic Portfolio Theory seeks to generate 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.

Date: 2024-11, Revised 2024-12
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