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Stochastic Analysis and Diffusion Processes$
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Stochastic Analysis and Diffusion Processes

Gopinath Kallianpur and P Sundar

Print publication date: 2014

Print ISBN-13: 9780199657063

Published to Oxford Scholarship Online: April 2014

DOI: 10.1093/acprof:oso/9780199657063.001.0001

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Stochastic Integration

Stochastic Integration

Chapter:
(p.90) 5 Stochastic Integration
Source:
Stochastic Analysis and Diffusion Processes
Author(s):

Gopinath Kallianpur

P. Sundar

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780199657063.003.0005

The Itô stochastic integral with respect to a Brownian motion is constructed, and its properties are shown in detail. The Itô formula is proved. Its applications include Lévy’s characterization of a Brownian motion, the Burkhölder-Davis-Gundy inequality, and the martingale representation theorem. Next, local times and the Tanaka formula are discussed. The Girsanov theorem on change of measures is proved in the last section.

Keywords:   stochastic integral, Itô, formula, Lévy’s characterization, Burkhölder-Davis-Gundy inequality, martingale representation theorem, local times, Tanaka formula, Girsanov theorem

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