Peter Kempthorne, MIT lecturer for 18.642 Topics in Mathematics with Applications in Finance, works through the foundations of stochastic calculus. The lecture builds Brownian motion with drift as a model for random processes, then constructs the Ito integral for both deterministic and random integrands, extending ordinary calculus to settings where functions jitter unpredictably. Kempthorne derives the Ito isometry, linking the variance of a stochastic integral to the norm of its integrand, and presents Ito's formula as the stochastic analogue of a Taylor expansion. He connects these tools to martingale problems and to partial differential equations that arise in quantitative finance, showing how the abstract machinery feeds directly into pricing and modeling questions. The session is dense with derivations on the board, aimed at students who already have a grounding in probability and calculus and are moving toward applications in financial mathematics.