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This monograph presents an advanced and unified treatment of four important issues that have dominated the theoretical research in mathematical finance for the last ten years: the fundamental theorem of asset pricing; utility maximization in incomplete markets; pricing in incomplete markets; and, the risk measurement of a static payoff and of a cash-flow stream. The powerful tools of convex analysis and duality theory are systematically applied to investigate these topics, under very general assumptions on the financial markets. This duality approach reveals the prominent role of the investor's preferences in all these fundamental issues and contributes to a deeper understanding of the economic aspects of the theory.