MTH5560 Partial differential equations in finance; From qualitative analysis to machine learning

This unit studies parabolic partial differential equations (PDEs) and their use in finance, progressing from a qualitative understanding of solution behaviour to machine learning methods for computing solutions. The qualitative phase interprets the operators that build these equations — the Laplacian as diffusion and the gradient as transport — and develops the maximum principle, the comparison theorem, and viscosity solutions, which reveal the crude behaviour of a solution without solving the equation. This understanding then guides the numerical phase: finite difference methods, the Monte-Carlo method, and machine learning approaches including the deep BSDE method and the random neural network method. The Black-Scholes-Merton equation for derivative pricing and the Hamilton-Jacobi-Bellman equation for stochastic control are derived as PDEs arising in finance. You will learn to read the qualitative structure of a financial PDE and pair that understanding with computational and machine learning solvers.