From Measures to It´ Integrals

By: Ekkehard Kopp

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Paperback | 31 March 2011

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From Measures to Ito Integrals gives a clear account of measure theory, leading via L2-theory to Brownian motion, Ito integrals and a brief look at martingale calculus. Modern probability theory and the applications of stochastic processes rely heavily on an understanding of basic measure theory. This text is ideal preparation for graduate-level courses in mathematical finance and perfect for any reader seeking a basic understanding of the mathematics underpinning the various applications of Ito calculus.

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Preface	p. vii
Probability and measure	p. 1
Do probabilists need measure theory?	p. I
Continuity of additive set functions	p. 1
Independent events	p. 8
Simple random walk	p. 10
Measures and distribution functions	p. 12
$-finite measures	p. 12
Generated $-fields and -systems	p. 15
Distribution functions	p. 17
Measurable functions/random variables	p. 19
Measurable real-valued functions	p. 19
Lebesgue- and Borel-measurable functions	p. 20
Stability properties	p. 21
Random variables and independence	p. 25
Integration and expectation	p. 30
Integrals of positive measurable functions	p. 31
The vector space L¹ of integrable functions	p. 36
Riemann v. Lebesgue integrals	p. 40
Product measures	p. 42
Calculating expectations	p. 44
L^p-spaces and conditional expectation	p. 49
L^p as a Banach space	p. 49
Orthogonal projections in L²	p. 54
Properties of conditional expectation	p. 56
Discrete-time martingales	p. 60
Discrete filtrations and martingales	p. 60
The Doob decomposition	p. 63
Discrete stochastic integrals	p. 65
Doob's inequalities	p. 67
Martingale convergence	p. 68
The Radon-Nikodym Theorem	p. 74
Brownian Motion	p. 78
Processes, paths and martingales	p. 78
Convergence of scaled random walks	p. 81
BM: construction and properties	p. 83
Martingale properties of BM	p. 88
Variation of BM	p. 92
Stochastic integrals	p. 95
The Ito integral	p. 95
The integral as a martingale	p. 101
Ito processes and the Ito formula	p. 104
The Black-Scholes model in finance	p. 108
Martingale calculus	p. 111
References	p. 118
Index	p. 119
Table of Contents provided by Ingram. All Rights Reserved.

From Measures to It´ Integrals

At a Glance

Paperback

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