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

2024-11-06

AI Summary

This paper addresses the problem of strong relative arbitrage in volatility-stabilized markets—i.e., constructing investment strategies that almost surely outperform a benchmark portfolio within a finite horizon. This is equivalent to finding the nonnegative minimal solution to a class of Cauchy-type partial differential equations characterized by non-uniqueness and high dimensionality. Methodologically, we innovatively introduce a time-changed Bessel bridge approach, integrated with weak solution theory and Monte Carlo path sampling, to design the first implementable continuous numerical algorithm that provably ensures both nonnegativity and minimality of the solution. Our method overcomes the dual bottlenecks of traditional approaches—lack of existence proofs and computational intractability. Numerical experiments on canonical volatility-stabilized market models demonstrate the algorithm’s stability, convergence, and strategic efficacy, significantly improving both the accuracy and practical applicability of optimal relative return strategies.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Economics, Online Markets and Human Computation: Sustainability of Web economics

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.

Problem

• Finding minimal nonnegative solutions for Cauchy PDEs in volatility-stabilized markets
• Addressing high dimensionality and non-uniqueness in PDE solutions for arbitrage
• Developing numerical methods using time-changed Bessel bridges for relative arbitrage

Innovation

• Uses time-changed Bessel bridges
• Solves Cauchy PDEs numerically
• Applies to volatility-stabilized markets