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This book develops a mathematical theory for finance, based on a
simple and intuitive absence-of-arbitrage principle. This posits
that it should not be possible to fund a non-trivial liability,
starting with initial capital arbitrarily near zero. The principle
is easy-to-test in specific models, as it is described in terms of
the underlying market characteristics; it is shown to be equivalent
to the existence of the so-called ""Kelly"" or growth-optimal
portfolio, of the log-optimal portfolio, and of appropriate local
martingale deflators. The resulting theory is powerful enough to
treat in great generality the fundamental questions of hedging,
valuation, and portfolio optimization. The book contains a
considerable amount of new research and results, as well as a
significant number of exercises. It can be used as a basic text for
graduate courses in Probability and Stochastic Analysis, and in
Mathematical Finance. No prior familiarity with finance is
required, but it is assumed that readers have a good working
knowledge of real analysis, measure theory, and of basic
probability theory. Familiarity with stochastic analysis is also
assumed, as is integration with respect to continuous
semimartingales.
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