Solving PDE in Finance

The aim of this note is to introduce some recent development in numerical solving of partial differential equations (PDE) with the use of deep neural networks. We particularly focus on PDE that arise in option pricing. In a first part, we introduce the notion of partial differential equation and give some example such equation in finance. In a second part, we briefly develop some classical numerical methods suach as Finite difference, its convergence and limits. Then we introduce the Galerkin method and its deep learning version whoch is a deterministic approach particularly adapted for high dimension PDE. In a last part, we introduce the probabilistic approach with Backward stochastic differential equation representation of PDE and a first approach to solve it with Neural Networks. Finally, we test these different approaches to option pricing problems.