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(Applied mathematics 17) Michael Meyer - Continuous stochastic calculus with app.pdf
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The current, prolonged boom in the US and European stock markets has increased
interest in the mathematics of security markets most notably the theory of stochastic
integration. Existing books on the subject seem to belong to one of two classes.
On the one hand there are rigorous accounts which develop the theory to great
depth without particular interest in finance and which make great demands on the
prerequisite knowledge and mathematical maturity of the reader. On the other hand
treatments which are aimed at application to finance are often of a nontechnical
nature providing the reader with little more than an ability to manipulate symbols to
which no meaning can be attached. The present book gives a rigorous development
of the theory of stochastic integration as it applies to the valuation of derivative
securities. It is hoped that a satisfactory balance between aesthetic appeal, degree
of generality, depth and ease of reading is achieved
Prerequisites are minimal. For the most part a basic knowledge of measure
theoretic probability and Hilbert space theory is sufficient. Slightly more advanced
functional analysis (Banach Alaoglu theorem) is used only once. The development
begins with the theory of discrete time martingales, in itself a charming subject.
From these humble origins we develop all the necessary tools to construct the
stochastic integral with respect to a general continuous semimartingale. The limitation
to continuous integrators greatly simplifies the exposition while still providing
a reasonable degree of generality. A leisurely pace is assumed throughout, proofs
are presented in complete detail and a certain amount of redundancy is maintained
in the writing, all with a view to make the reading as effortless and enjoyable as
possible.
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