The lecture provides an in-depth introduction to stochastic calculus, focusing on Brownian motion with drift and the construction of Itô integrals, which extend ordinary calculus to stochastic processes. Key concepts include the definition of Itô integrals for random and deterministic functions, the Itô isometry connecting variance and integrand norms, and Itô’s formula, which generalizes Taylor expansions to stochastic settings, enabling applications such as solving partial differential equations and martingale problems in quantitative finance.