Peter Kempthorne continues his MIT course 18.642, Topics in Mathematics with Applications in Finance, with a second lecture on stochastic processes focused on Brownian motion. He develops the process as a model with continuous, independent increments whose variance grows linearly with time, then works through its core properties: the Markov property, the reflection principle, and quadratic variation. The lecture extends the basic model to Brownian motion with drift, reflected and absorbed Brownian motion, and the Brownian bridge, connecting each variant to problems in derivative pricing. Running eighty one minutes, the session builds the probabilistic machinery that underlies continuous-time finance models, assuming familiarity with the first stochastic processes lecture in the series. Kempthorne works through definitions and derivations at the board, aimed at students who need these tools for option pricing and risk models later in the course.