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

TL;DR

This paper develops numerical methods to solve high-dimensional, non-unique PDEs related to relative arbitrage in volatility-stabilized markets, enabling better investment strategies.

Contribution

It introduces a practical algorithm using time-changed Bessel bridges to numerically solve the PDE for relative arbitrage in complex market models.

Findings

Numerical methods effectively solve the PDE in volatility-stabilized markets.

The approach demonstrates promising results in example scenarios.

The method addresses high-dimensional and non-unique solution challenges.

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.