Stochastic Analysis
Seiten
2020
|
1st ed. 2020
Springer Verlag, Singapore
978-981-15-8863-1 (ISBN)
Springer Verlag, Singapore
978-981-15-8863-1 (ISBN)
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.
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.
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.
The author is currently Professor Emeritus at The University of Tokyo and visiting Professor at Meiji University. He previously held positions at The University of Tokyo and Research Institute for Mathematical Sciences, Kyoto University. He was an invited speaker at the ICM 1990.
Chapter 1. Preparations from probability theory.- Chapter 2. Martingale with discrete parameter.- Chapter 3. Martingale with continuous parameter.- Chapter 4. Stochastic integral.- Chapter 5. Applications of stochastic integral.- Chapter 6. Stochastic differential equation.- Chapter 7. Application to finance.- Chapter 8. Appendices.- References.
Erscheinungsdatum | 26.10.2020 |
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Reihe/Serie | Monographs in Mathematical Economics ; 3 |
Zusatzinfo | 1 Illustrations, black and white; XII, 218 p. 1 illus. |
Verlagsort | Singapore |
Sprache | englisch |
Maße | 155 x 235 mm |
Themenwelt | Mathematik / Informatik ► Mathematik ► Analysis |
Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik | |
ISBN-10 | 981-15-8863-5 / 9811588635 |
ISBN-13 | 978-981-15-8863-1 / 9789811588631 |
Zustand | Neuware |
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