Stochastic Processes

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;
>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.