Paper: arXiv 2411.13558

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.

Complexity vs Empirical Score

  • Math Complexity: 9.5/10
  • Empirical Rigor: 4.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: 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.

Research Flowchart

  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