Pathwise Portfolio Theory and Market Viability
Ioannis Karatzas, Donghan Kim
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.
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