This final lecture provides an in-depth discussion of Itô’s formula and its generalizations, illustrating how it applies to functions of Brownian motion, especially in finance for modeling derivative pricing and geometric Brownian motion. It also explains the derivation of the Black-Scholes differential equation through risk-neutral hedging, the solution of the heat (diffusion) equation as a fundamental tool for solving such PDEs, and introduces more advanced stochastic differential equations like the Ornstein-Uhlenbeck process, emphasizing their broad applications beyond finance.