Stochastic Analysis

This book covers probability theory, discrete-time martingales, continuous time square integrable martingales, stochastic integrations, and applications to mathematical finance. It requires preliminary knowledge of linear algebra and measure theory, and rigorous proofs are provided for theorems, propositions, and lemmas.

Format: Paperback / softback
Length: 218 pages
Publication date: 21 October 2021
Publisher: Springer Verlag, Singapore

This comprehensive textbook is meticulously crafted for advanced university students pursuing majors in probability theory or mathematical finance. It commences with a thorough review of probability theory concepts, encompassing fundamental results and their applications. The subsequent chapters delve into discrete-time martingales, continuous time square integrable martingales, stochastic integrations with respect to continuous local martingales, and stochastic differential equations driven by Brownian motions. The final chapter presents practical applications of these mathematical frameworks in the realm of mathematical finance.

To fully engage with the content, the reader is expected to possess a solid foundation in linear algebra and measure theory. Extensive proofs accompany each theorem, proposition, and lemma, ensuring the accuracy and rigor of the material presented.

One distinctive aspect of this textbook is its approach to conditional expectations. While other textbooks may follow a different definition, here, a slight variation is employed. Specifically, for the Doob–Meyer decomposition theorem, square integrable submartingales are solely considered, and elementary facts of square integrable functions are utilized in the proof. In the realm of stochastic differential equations, the Euler–Maruyama approximation plays a pivotal role, primarily for proving the uniqueness of martingale problems and the smoothness of solutions of stochastic differential equations.

By offering a comprehensive and rigorous treatment of these topics, this textbook serves as an invaluable resource for advanced students seeking to deepen their understanding of probability theory and mathematical finance. Its clear explanations, detailed proofs, and extensive examples make it suitable for both self-study and classroom instruction. Whether you are a aspiring researcher or a professional seeking to expand your expertise in these fields, this book will undoubtedly be a valuable companion on your academic journey.

Weight: 361g
Dimension: 235 x 155 (mm)
ISBN-13: 9789811588662
Edition number: 1st ed. 2020