内容简介 · · · · · ·
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.
Author(s): Ekkehard Kopp
Publisher: Cambridge University Press
Year: 2011
Language: English
Pages: 128
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Contents
Preface
1 Probability and measure
1.1 Do probabilists need measure theory?
1.2 Continuity of additive set functions
1.3 Independent events
1.4 Simple random walk
2 Measures and distribution functions
2.1 σ-finite measures
2.2 Generated σ-fields and π-systems
2.3 Distribution functions
3 Measurable functions/random variables
3.1 Measurable real-valued functions
3.2 Lebesgue- and Borel-measurable functions
3.3 Stability properties
3.4 Random variables and independence
4 Integration and expectation
4.1 Integrals of positive measurable functions
4.2 The vector space L1 of integrable functions
4.3 Riemann v. Lebesgue integrals
4.4 Product measures
4.5 Calculating expectations
5 Lp-spaces and conditional expectation
5.1 Lp as a Banach space
5.2 Orthogonal projections in L2
5.3 Properties of conditional expectation
6 Discrete-time martingales
6.1 Discrete filtrations and martingales
6.2 The Doob decomposition
6.3 Discrete stochastic integrals
6.4 Doob's inequalities
6.5 Martingale convergence
6.6 The Radon–Nikodym Theorem
7 Brownian Motion
7.1 Processes, paths and martingales
7.2 Convergence of scaled random walks
7.3 BM: construction and properties
7.4 Martingale properties of BM
7.5 Variation of BM
8 Stochastic integrals
8.1 The Itô integral
8.2 The integral as a martingale
8.3 Itô processes and the Itô formula
8.4 The Black–Scholes model in finance
8.5 Martingale calculus
References
Index
