Pathwise Portfolio Theory and Market Viability

2026-07-21

AI Summary

This work develops a portfolio theory within a purely path-dependent framework that dispenses with probabilistic assumptions, investigating the interplay among market feasibility, growth optimality, and numéraire properties. By introducing trend extractors and their residual paths as substitutes for semimartingale decomposition, and integrating Föllmer’s pathwise integral with pathwise calculus, the paper establishes a generalized toolkit for path-dependent analysis. Its central contribution lies in rigorously disentangling and characterizing—within this setting—the equivalence between growth optimality and the numéraire property from the equivalence between feasibility and boundedness, demonstrating that these two equivalences may belong to distinct equivalence classes. The resulting theory not only encompasses the semimartingale case as a special instance but also demonstrates its scope and extensibility through two non-degenerate examples.

Abstract

The theory of portfolios, and its allied notions and fundamental results concerning growth optimality, the numéraire property, and ``market viability'' -- which rules out the possibility of financing nontrivial future liability streams starting with arbitrarily small initial capital -- is developed in a pathwise setting, completely devoid of probabilistic considerations. The approach replaces the familiar semimartingale decomposition of stochastic analysis for assets' returns, by decompositions generated through suitable trend extractors and their associated residual paths; then deploys Föllmer's celebrated pathwise version of classical Itô integration and calculus. The resulting growth-numéraire and viability-boundedness equivalences bear considerable similarities to their semimartingale counterparts, but need not collapse into a single equivalence class in the pathwise setting; this separation is illustrated by two examples.

Problem

• pathwise portfolio theory
• market viability
• numéraire property
• growth optimality
• non-probabilistic finance

Innovation

• pathwise analysis
• trend extraction
• numéraire property
• market viability
• Föllmer integration