Stochastic Calculus Review

The FE-610 review consolidates the first half of the course: probability spaces and filtrations, martingales, Brownian motion, quadratic variation, the Itô integral, Itô's formula, and the Black–Scholes hedge.

The compact rules worth carrying forward are

$$dW_t\,dt=0,\qquad (dW_t)^2=dt,\qquad (dt)^2=0.$$

They are a mnemonic for the limiting variation calculation, not ordinary algebra. The review also separates three questions that are easy to conflate: what is measurable now, which process is a martingale under the current measure, and which payoff is being priced.