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.Contact Emmanuel MAITRE
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
Mathematical analysis (normed spaces, elementary Fourier analysis), linear algebra, basic numerical methods.
An exam at the end of the term (E).
N1 = Examen écrit session 1
N2 = Examen écrit session 2 ou oral
L.C. Evans : Partial differential equations (AMS)
D.P. Bertsekas : Dynamic programming and optimal control (MIT)