Pathwise analysis of log-optimal portfolios

TL;DR

This paper develops a pathwise, deterministic framework using rough path theory to analyze the stability and approximation of log-optimal portfolios along individual financial market trajectories.

Contribution

It introduces a novel pathwise approach to construct and analyze log-optimal portfolios, providing stability and error estimates without relying on probabilistic assumptions.

Findings

Established pathwise stability estimates for log-optimal portfolios.

Derived error bounds for time-discretization of portfolios.

Provided a deterministic framework applicable to individual market paths.

Abstract

Based on the theory of c\`adl\`ag 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 It\^o 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.