Stochastic Processes

In Monte Carlo you manufactured tomorrow by rolling dice ten thousand times — but something always told those dice what to do. Stochastic processes are where you stop trusting that little drift-and-volatility recipe and start understanding it: Monte Carlo rolls the dice; stochastic processes are the rulebook for which dice, why those dice, and what guarantees they buy you.

Climbing one rung at a time, this topic covers:

• What a stochastic process is — a random variable that unfolds in time, the difference between a single sample path and the cross-sectional distribution, filtrations (growing information), stationarity and independent increments.
• Random walks & martingales — the coin-flip walk, what drift adds, and the martingale (best guess for tomorrow is exactly today) as the fingerprint of an efficient market.
• Markov chains — the memoryless property, transition matrices, n-step transitions, stationary distributions, and calm-vs-crisis regimes as a chain in disguise.
• Brownian motion — the scaling limit of infinitely many infinitesimal steps: Gaussian independent increments, continuous-but-nowhere-differentiable paths, and the strange quadratic variation that seeds stochastic calculus.
• Itô’s lemma & SDEs — drift plus diffusion, why ordinary calculus breaks on a Brownian path, where the $\tfrac{1}{2}\sigma^2$ correction comes from, and how solving the SDE yields geometric Brownian motion and lognormal prices.
• Mean reversion & jumps — the Ornstein–Uhlenbeck process (with a half-life you read off the parameters when $\kappa > 0$), the Merton jump-diffusion for fat tails and overnight crash risk, and how to choose between models for a real series.

It closes with a graded exam. Master this and you’ve learned the native language of quantitative finance — every option-pricing model, term-structure model, and risk engine from here on is written in stochastic differential equations, and the rest of the field stops being magic and starts being grammar.