The central mathematical concept in the theory of frictionless market is a martingale measure. The authors argue that for financial markets with proportional transaction costs this concept should be replaced by the concept of consistent price system which is a martingale evolving in the duals to the solvency cones. The book presents a unified treatment of various problems arising in the theory of financial markets with friction. It gives a succinct account of arbitrage theory for financial markets with and without transaction costs based on a synthesis of ideas from the finite-dimensional geometry, functional analysis, and stochastic processes. For practitioners working with low-liquid markets the chapter on Lelands approximate hedging strategies will be of especial interest.The book is supplemented by an appendix that provides a toolbox containing auxiliary results from various branches of mathematics used in the proofs.

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