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. It thoroughly covers essential tools of stochastic calculus, such as Itô’s formula, the optional stopping theorem, and Girsanov’s theorem, accompanied by numerous illustrative examples. An introduction to Markov processes is included, highlighting applications to stochastic differential equations and the relationship between Brownian motion and partial differential equations. The final chapter discusses the theory of local times of semimartingales. Since its inception by Itô, stochastic calculus has become a crucial technique in modern probability theory, influencing both theoretical advancements and practical applications, particularly in mathematical finance. This work provides a strong theoretical foundation for readers interested in these developments. It is tailored for beginning graduate or advanced undergraduate students, emphasizing concise and efficient presentation without sacrificing mathematical rigor. The author has taught this 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.