This thoroughly updated second edition offers a unified, modern pathway from the Kolmogorov foundations of probability to the tools of stochastic calculus—and on to applications in finance, statistics, and risk. With clarity and breadth, it develops martingale and semimartingale theory alongside stochastic differential equations, keeping both discrete- and continuous-time viewpoints in play.

What’s new in the 2nd Edition

• Optional Stochastic Analysis on non-“usual” filtrations: the first textbook presentation of optional processes on stochastic bases beyond the standard right-continuous, complete setting, with an accompanying optional stochastic calculus.
• Optional SDEs and stochastic exponentials/logarithms: existence-uniqueness theory and product/inverse rules, with financial modeling worked out in this optional-semimartingale framework.
• New applications: Stochastic Regression Analysis and Risk Theory, showing how optional tools yield estimation results and ruin-probability bounds in general settings.
• Expanded exercises with solutions: a substantially enlarged Supplement (Ch. 15) featuring problems that reinforce both core theory and applications.

Designed for senior undergraduates, graduate students, and instructors, the book also serves researchers and practitioners who need a concise, example-driven route from measure-theoretic probability to the techniques used in finance, statistics, and risk modeling. Abundant worked examples and a comprehensive set of problems—with hints and solutions—make it ideal for self study or course adoption.

Alexander Melnikov is a professor at the University of Alberta working in stochastic analysis and its applications in finance, statistics, and insurance. He is the author of nine books and about one hundred fifty papers in leading academic journals and venues. He is a fellow of the Russian Academy of Natural Sciences, a recipient of the Leontiev medal of this academy and the McCalla Professorship of the University of Alberta. In addition to his academic engagements, he held several senior positions in business and professional organizations: chief-scientist at Risk-Invest Deutschland (Frankfurt), vice-president of the Russian Society of Actuaries, deputy director at the Center for Actuarial and Financial Studies (Moscow), and senior research consultant at the Model Capital Management (Boston).