- Lecturer: Nazem Khan
- Lecturer: Justin Sirignano
Course term: Michaelmas
Course lecture information: 5 hours of lectures in week -1.
Course overview:
This course reviews the basic theory of partial differential equations (PDEs). Topics covered include linear transport equations, the heat equation, and the Black-Scholes equation. Connections to stochastic differential equations and finance are highlighted. Mathematical techniques for solving PDEs (separation of variables, Fourier transform) are presented. Finite-difference methods for the numerical solution of PDEs will also be discussed.
Course synopsis:
• Linear transport equation
• Heat Equation
• Separation of Variables
• Fourier transform for solving the heat equation
• Fourier transform for solving the multi-dimensional heat equation
• Connection between the heat equation and Brownian motion (i.e., the solution to the heat equation is an expectation of a function of a Brownian motion)
• Maximum principle and uniqueness for the heat equation
• Energy method for proving uniqueness
• Black-Scholes PDE
• Derivation of finite-difference method and implementation in Python for numerically solving the heat equation
• Heat Equation
• Separation of Variables
• Fourier transform for solving the heat equation
• Fourier transform for solving the multi-dimensional heat equation
• Connection between the heat equation and Brownian motion (i.e., the solution to the heat equation is an expectation of a function of a Brownian motion)
• Maximum principle and uniqueness for the heat equation
• Energy method for proving uniqueness
• Black-Scholes PDE
• Derivation of finite-difference method and implementation in Python for numerically solving the heat equation