Lecture 24
Video description: 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.
Stochastic Calculus Slides (PDF)
Simulations of Brownian Motion (PDF)
Lecture 25
Video description: This final lecture provides an in-depth discussion of Itô’s formula and its generalizations, illustrating how it applies to functions of Brownian motion, especially in finance for modeling derivative pricing and geometric Brownian motion. It also explains the derivation of the Black-Scholes differential equation through risk-neutral hedging, the solution of the heat (diffusion) equation as a fundamental tool for solving such PDEs, and introduces more advanced stochastic differential equations like the Ornstein-Uhlenbeck process, emphasizing their broad applications beyond finance.
