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

ArXiv ID: 2411.13558 “View on arXiv”

Authors: Unknown

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: Stochastic Portfolio Theory, Relative Arbitrage, Partial Differential Equations (PDE), Time-changed Bessel Bridges, Volatility-stabilized Markets, Equity

Complexity vs Empirical Score

• Math Complexity: 9.5/10
• Empirical Rigor: 4.0/10
• Quadrant: Lab Rats
• Why: The paper is dense with advanced mathematics including high-dimensional PDEs, Cauchy problems, and stochastic differential equations, but the empirical component is limited to numerical simulations without real-world data or backtesting.

  flowchart TD
    Goal["Research Goal:<br>Find nonnegative minimal solution for Cauchy PDE<br>in volatility-stabilized markets"]

    Data["Input Data:<br>Volatility-stabilized market parameters<br>Equity price dynamics<br>Time horizon T"]

    Method["Methodology:<br>Time-changed Bessel Bridges<br>Transform PDE into heat equation"]

    Process["Computational Process:<br>Solve transformed PDE numerically<br>Apply inverse time-change"]

    Outcome["Key Outcome:<br>Identification of smallest nonnegative solution<br>Characterizes highest relative return<br>Practical numerical algorithm"]

    Goal --> Data
    Data --> Method
    Method --> Process
    Process --> Outcome