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Explore the mathematical foundations of stochastic calculus in this advanced lecture from MIT's Topics in Mathematics with Applications in Finance course. Delve into Brownian motion with drift and learn how to construct Itô integrals that extend ordinary calculus to handle stochastic processes. Master key concepts including the definition of Itô integrals for both random and deterministic functions, understand the Itô isometry that connects variance with integrand norms, and discover Itô's formula, which serves as the stochastic equivalent of Taylor expansions. Apply these powerful mathematical tools to solve partial differential equations and martingale problems commonly encountered in quantitative finance, gaining essential skills for advanced mathematical modeling in financial markets.
