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

Overview

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.

Offerings

Assessment

The Handbook lists an examination for this unit.

Assessment details may change. Please refer to the assessment information in Moodle closer to the start of the teaching period.

Requisites

Learning outcomes

1. Articulate specialised mathematical concepts within the field of partial differential equations;
2. Recognise the complex connections between stochastic analysis and partial differential equations;
3. Apply sophisticated mathematical modelling skills to problems in partial differential equations that relate to financial markets;
4. Demonstrate critical thinking and problem solving skills within the context of financial mathematics;
5. Formulate expert solutions, both analytical and numerical, to practical financial problems using specialised cognitive and technical skills within the field of partial differential equations;
6. Communicate complex information in an accessible format to a non-mathematical audience;
7. Apply machine learning methods, including the deep BSDE and random neural network methods, to compute solutions of partial differential equations arising in finance, drawing on qualitative analysis of solution behaviour to inform the numerical approach.

Workload

• 3 hours of seminars; • One hour of applied classes and • 8 hours of independent study per week (including working on assessments and revision)

Student experience, not official rules. Nothing here changes what the Handbook says.