Finding the nonnegative minimal solutions of Cauchy PDEs in a volatility-stabilized market
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.
Keywords
Cite
@article{arxiv.2411.13558,
title = {Finding the nonnegative minimal solutions of Cauchy PDEs in a volatility-stabilized market},
author = {Nicole Tianjiao Yang and Tomoyuki Ichiba},
journal= {arXiv preprint arXiv:2411.13558},
year = {2025}
}