Authors:
- Defines conditional exceptions differently than in other books
- Uses only elementary facts for proof of the Doob–Meyer decomposition theorem for special cases
- Shows how the Euler–Maruyama approximation plays an important role in proving the uniqueness of martingale problems
Part of the book series: Monographs in Mathematical Economics (MOME, volume 3)
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Table of contents (8 chapters)
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Front Matter
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Back Matter
About this book
This book is intended for university seniors and graduate students majoring in probability theory or mathematical finance. In the first chapter, results in probability theory are reviewed. Then, it follows a discussion of discrete-time martingales, continuous time square integrable martingales (particularly, continuous martingales of continuous paths), stochastic integrations with respect to continuous local martingales, and stochastic differential equations driven by Brownian motions. In the final chapter, applications to mathematical finance are given. The preliminary knowledge needed by the reader is linear algebra and measure theory. Rigorous proofs are provided for theorems, propositions, and lemmas.
In this book, the definition of conditional expectations is slightly different than what is usually found in other textbooks. For the Doob–Meyer decomposition theorem, only square integrable submartingales are considered, and only elementary facts ofthe square integrable functions are used in the proof. In stochastic differential equations, the Euler–Maruyama approximation is used mainly to prove the uniqueness of martingale problems and the smoothness of solutions of stochastic differential equations.
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Authors and Affiliations
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Graduate School of Mathematical Sciences, The University of Tokyo, Tokyo, Japan
Shigeo Kusuoka
About the author
Bibliographic Information
Book Title: Stochastic Analysis
Authors: Shigeo Kusuoka
Series Title: Monographs in Mathematical Economics
DOI: https://doi.org/10.1007/978-981-15-8864-8
Publisher: Springer Singapore
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020
Hardcover ISBN: 978-981-15-8863-1Published: 20 October 2020
Softcover ISBN: 978-981-15-8866-2Published: 21 October 2021
eBook ISBN: 978-981-15-8864-8Published: 20 October 2020
Series ISSN: 2364-8279
Series E-ISSN: 2364-8287
Edition Number: 1
Number of Pages: XII, 218
Number of Illustrations: 1 b/w illustrations
Topics: Probability Theory and Stochastic Processes, Measure and Integration, Functional Analysis, Applications of Mathematics