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Informatique et Mathématiques appliquées  # Partial differential equations for finance - WMMFMA20

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• #### Number of hours

• Lectures : 15.0
• Tutorials : -
• Laboratory works : 3.0
• Projects : -
• Internship : -
• Written tests : -
ECTS : 2.0
• Officials : Emmanuel MAITRE

### Goals

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.

Content

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.

Tests

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

N1 = Examen écrit session 1
N2 = Examen écrit session 2 ou oral

Calendar

The course exists in the following branches:

• Curriculum - Financial Engineering - Semester 9
see the course schedule for 2019-2020

Course ID : WMMFMA20
Course language(s): The course is attached to the following structures:

You can find this course among all other courses.

Bibliography

L.C. Evans : Partial differential equations (AMS)
D.P. Bertsekas : Dynamic programming and optimal control (MIT)

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Date of update January 15, 2017 École nationale supérieure d'informatique et de mathématiques appliquées
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