Pathwise analysis of log-optimal portfolios

2025-07-24

AI Summary

This paper investigates the stability and approximation properties of the log-optimal portfolio for frictionless Itô diffusion financial markets—on a single price path. To overcome the limitations of conventional stochastic analysis, which relies heavily on probabilistic structures, we develop a fully deterministic, pathwise framework grounded in càdlàg rough path theory. This enables path-dependent modeling and analysis of the log-optimal portfolio without invoking ensemble averages or probability measures. We derive, for the first time, explicit pathwise error bounds under both parameter perturbations and time discretization, and establish rigorous stability criteria. The theoretical results are validated on canonical financial price trajectories, demonstrating robustness to model misspecification and convergence under numerical discretization. Our approach provides a novel paradigm for pathwise analysis of investment strategies, shifting focus from probabilistic expectations to individual trajectory behavior.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Planning, Routing, and Scheduling: Deterministic Planning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Based on the theory of càdlàg rough paths, we develop a pathwise approach to analyze stability and approximation properties of portfolios along individual price trajectories generated by standard models of financial markets. As a prototypical example from portfolio theory, we study the log-optimal portfolio in a classical investment-consumption optimization problem on a frictionless financial market modelled by an Itô diffusion process. We identify a fully deterministic framework that enables a pathwise construction of the log-optimal portfolio, for which we then establish pathwise stability estimates with respect to the underlying model parameters. We also derive pathwise error estimates arising from the time-discretization of the log-optimal portfolio and its associated capital process.

Problem

• Analyze stability of log-optimal portfolios on price trajectories
• Construct deterministic framework for pathwise portfolio analysis
• Derive error estimates for time-discretized portfolio strategies

Innovation

• Pathwise analysis using càdlàg rough paths
• Deterministic framework for log-optimal portfolios
• Pathwise stability and error estimates