Peter Kempthorne closes MIT's 18.642 Topics in Mathematics with Applications in Finance with a detailed treatment of Ito's formula and its generalizations, showing how it governs functions of Brownian motion and underlies geometric Brownian motion models used in finance. He derives the Black-Scholes differential equation through risk-neutral hedging arguments, then shows how solving the heat (diffusion) equation provides the key analytic tool for such PDEs. The lecture ends by introducing more advanced stochastic differential equations, including the Ornstein-Uhlenbeck process, and points out their uses beyond derivative pricing. Delivered as a chalkboard-and-slides lecture at the standard MIT OpenCourseWare production level, running 81 minutes, it assumes prior familiarity with stochastic calculus from earlier sessions in the course.