Stochastic Calculus and Brownian Motion

"Stochastic Calculus and Brownian Motion" is a comprehensive guide crafted for students and professionals in mathematical sciences, focusing on stochastic processes and their real-world applications in finance, physics, and engineering. We explore key concepts and mathematical foundations of random movements and their practical implications. At its core, the book delves into Brownian motion, the random movement of particles suspended in a fluid, as described by Robert Brown in the 19th century. This phenomenon forms a cornerstone of modern probability theory and serves as a model for randomness in physical systems and financial models describing stock market behaviors. We also cover martingales, mathematical sequences where future values depend on present values, akin to a fair game in gambling. The book demonstrates how martingales are used to model stochastic processes and their calibration in real-world scenarios. Stochastic calculus extends these ideas into continuous time, integrating calculus with random processes. Our guide provides the tools to understand and apply Itô calculus, crucial for advanced financial models like pricing derivatives and managing risks. Written clearly and systematically, the book includes examples and exercises to reinforce concepts and showcase their real-world applications. It serves as an invaluable resource for students, educators, and professionals globally.

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E-kirja: 20. helmikuuta 2025