Pathwise Portfolio Theory and Market Viability

The theory of portfolios, and its allied notions and fundamental results concerning growth optimality, the num\'eraire 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 F\"ollmer's celebrated pathwise version of classical It\^o integration and calculus. The resulting growth-num\'eraire 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.