# Partial differential equations for finance - WMMFMA26

• #### Number of hours

• Lectures 12.0
• Projects -
• Tutorials -
• Internship -
• Laboratory works 6.0
• Written tests -

ECTS 2.0

## Goal(s)

Introduction to the different classes of partial differential equations: elliptic, parabolic and hyperbolic. Their characteristics and what this implies in terms of numerical approximations. Applications to various PDE models involved in mathematical finance. Among those, we will consider the Black-Scholes equation, Hamilton-Jacobi equations, in the framework of dynamic optimal control theory.

Responsible(s)

Olivier ZAHM

## Content(s)

1. Introduction: origin of partial differential equations (PDE) in mathematical finance
2. Different types of partial differential equations: parabolic, elliptic, hyperbolic and of mixed type
What are the physical phenomenon associated to, and how do they appear in e.g. Black-Scholes equation (diffusion part, transport part).
3. Partial differential equations, initial and boundary conditions: how to set them?
Notion of characteristic surface for a PDE.
4. Hamilton-Jacobi equations and introduction to dynamic optimal control
5. Some elements of numerical analysis of PDEs: theory and practice

Prerequisites

Mathematical analysis (normed spaces, elementary Fourier analysis), linear algebra, basic numerical methods.

Test

An exam at the end of the term (E).

• MCC en présentiel **
N1 = 1/2 TP encadré + 1/2 examen écrit
N2 = 1/2 TP encadré + 1/2 examen écrit
• MCC en distanciel **
N1 = 1/2 TP à distance + 1/2 devoir à la maison
N2 = devoir à la maison

Calendar

The course exists in the following branches:

• Curriculum - Financial Engineering - Semester 9
see the course schedule for 2023-2024