Growth-optimal investments and numeraire portfolios under transaction costs: An analysis based on the von Neumann-Gale model

The aim of this work is to extend the capital growth theory developed by Kelly, Breiman, Cover and others to asset market models with transaction costs. We de…ne a natural generalization of the notion of a numeraire portfolio proposed by Long and show how such portfolios can be used for constructing growth-optimal investment strategies. The analysis is based on the classical von Neumann-Gale model of economic dynamics, a stochastic version of which we use as a framework for the modelling of …nancial markets with frictions. Key words: capital growth theory, transaction costs, numeraire portfolios, random dynamical systems, convex multivalued operators, von Neumann–Gale model, rapid paths 2000 Mathematics Subject Classi…cation: 37H99, 37H15, 91B62, 91B28. JEL-Classi…cation: C61, C62, O41, G10. a School of Mathematics, Loughborough University, Leicestershire, LE11 3TU, UK. b Economics Department, University of Manchester, Oxford Road, Manchester M13 9PL, UK. E-mail: [email protected]. Phone: 0161-2754275. Fax: 0161-2754812. (Corresponding author.) Mathematics Department, University of Missouri, Columbia, MO 65211, USA.