Stochastic Processes: From Brownian Motion to Markets

A compact, mathematically serious introduction to stochastic processes through probability, physics, computation, and finance.

Stochastic Processes: From Brownian Motion to Markets develops randomness as structure. Beginning with the simple one-dimensional random walk, the book builds toward Brownian motion, diffusion equations, Langevin dynamics, heavy-tailed models, option pricing, model risk, and computational validation. Rather than presenting stochastic processes as a collection of formulas, this book emphasizes the modeling questions behind them: What is random? How does uncertainty evolve? What can be computed? And where does the model fail? Readers will learn how to:

• derive random-walk distributions from path counting;
• understand diffusion scaling and the Gaussian limit;
• connect Brownian motion to the heat equation and Langevin dynamics;
• recognize heavy tails, Lévy flights, and anomalous diffusion;
• interpret Black-Scholes, risk-neutral valuation, and model failure;
• use simulation projects to test and visualize stochastic models. Written for students with calculus, basic probability, and algebra. This book is especially suited for readers interested in mathematical physics, quantitative finance, stochastic modeling, or computational approaches to uncertainty.