Stochastic Calculus and Stochastic Differential Equations - Lecture 25

Explore advanced stochastic calculus concepts and their applications in mathematical finance through this comprehensive lecture from MIT's Topics in Mathematics with Applications in Finance course. Delve into Itô's formula and its generalizations, examining how this fundamental tool applies to functions of Brownian motion with particular emphasis on derivative pricing models and geometric Brownian motion in financial contexts. Learn the systematic derivation of the Black-Scholes differential equation through risk-neutral hedging principles, and master the solution techniques for the heat (diffusion) equation as an essential method for solving partial differential equations in finance. Discover advanced stochastic differential equations including the Ornstein-Uhlenbeck process, understanding their mathematical properties and recognizing their broad applications that extend well beyond financial modeling into various fields of applied mathematics and physics.