Brownian Motion, Martingales, and Stochastic Calculus

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This book presents a rigorous and self-contained exploration of stochastic integration and calculus within the framework of continuous semimartingales. Key tools such as Itô’s formula, the optional stopping theorem, and Girsanov’s theorem are discussed in detail, accompanied by numerous illustrative examples. An introduction to Markov processes is included, highlighting applications to stochastic differential equations and the connections between Brownian motion and partial differential equations. The final chapter addresses the theory of local times of semimartingales. Since Itô's invention, stochastic calculus has become a crucial technique in modern probability theory, influencing both theoretical advancements and practical applications in fields like mathematical finance. This work provides a strong theoretical foundation for readers interested in these developments. It is particularly beneficial for beginning graduate or advanced undergraduate students, emphasizing a concise and efficient presentation while maintaining mathematical rigor. The author has taught the material in graduate courses at prestigious French universities, and the detailed proofs make it suitable for self-study. Additionally, numerous exercises help readers familiarize themselves with the tools of stochastic calculus.