From Measures to Ito Integrals

by Kopp, Ekkehard

• ISBN13: 9781107400863
• ISBN10: 1107400864
• Format: Paperback
• Copyright: 2011-05-09
• Publisher: Cambridge Univ Pr

Summary

From Measures to Itô Integrals gives a clear account of measure theory, leading via L2-theory to Brownian motion, Itô 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 Itô calculus.

Author Biography

Ekkehard Kopp studied at Stellenbosch University and obtained has D. Phil, from the University of Oxford in 1973. He held academic positions at the University of Hull from 1970 until his retirement in 2004, including serving as Dean of Mathematics and Pro-Vice-Chancellor. He is the author of over 50 publications in analysis, probability and Mathematical finance.

Table of Contents

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
Lp-spaces and conditional expectation	p. 49
Lp 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
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