Exercises and Solutions in Stochastic Calculus: Stochastic Calculus

This collection of solved exercises offers an in‑depth exploration of stochastic calculus, designed for master's students and researchers. It covers the foundations of Brownian motion, martingales, Itô stochastic integrals, and stochastic differential equations (SDEs). The book is structured around fifty carefully selected exercises, grouped by theme, that progressively guide the reader from basic concepts to advanced topics. The exercises are organised into several parts: Brownian motion and its properties (path properties, quadratic variation, the law of the iterated logarithm, and first passage times); martingales in continuous time (exponential martingales, Girsanov's theorem, and change of measure); stochastic integration and Itô calculus (construction of the Itô integral, Itô's formula, and its applications); stochastic differential equations (existence and uniqueness, linear SDEs, comparison theorems, and numerical methods); and applications in finance (Black–Scholes model, option pricing), physics (Bessel processes, Langevin dynamics), and biology (population models). Each exercise is accompanied by a \textbf that explains the reasoning step by step, highlights common pitfalls, and provides intuitive interpretations of the results. Many solutions also include \textbf to illustrate the theoretical findings and to give the reader hands‑on experience with numerical stochastic calculus. To support self‑study, the book includes two appendices: \textit (Appendix~A) reviews the basic rules of Itô calculus and the derivation of key formulas, while \textit (Appendix~B) provides a quick reference for frequently encountered integrals. These appendices make the book self‑contained and serve as a convenient resource for both students and practitioners. This work is intended to be a companion for advanced courses in stochastic calculus and for researchers who wish to master the tools of the field. The progressive nature of the exercises and the thorough solutions enable the reader to develop an autonomous understanding, while the Python simulations bridge the gap between theory and practice. Read more