# Lectures on BSDE

# Lectures on Backward Stochastic Differential Equations by Alexandre Popier, Le Mans Université

### 14-16 October 2019, METU, Bilkent, Ankara, Turkey

- Lecture 1: From Martingales, PDE, Maximum principle to BSDE | October 14, 2019, 14:40 | METU, Department of Mathematics | M203 | Slides
- Seminar: BSDE with singular terminal conditions: how to handle the explosion of the solution | October 15, 15:40 | METU, Institute of Applied Mathematics | Seminar Room | Slides
- Lecture 2: BSDE: extensions, an application in financial economics | October 16, 15:40 | Bilkent, Department of Industrial Engineering | EA409 | Slides

Detailed content for each event is below. These lectures are a part of the BSDE with Singular Terminal Values 2018-2020-Tubitak 1001 Project.

## Lecture 1: From Martingales, PDE, Maximum principle to BSDE | October 14, Monday, 14:40 | METU, Department of Mathematics | M203

**October 14, 2019, 14:40, METU, Department of Mathematics, M203**

**Motivation and introduction**

- Ordinary differential equations and martingales.
- Feynman–Kac formula : from linear to semilinear partial differential equations.
- Optimal control problem and stochastic Pontryagin maximum principle.

**Results in the BSDE theory**

- Existence and uniqueness of the solution in the Lipschitz and quadratic setting.
- Linear BSDEs and the comparison principle.
- The monotone and L p setting.
- Random horizon.

**Some words about numerics**

## Seminar: BSDE with singular terminal conditions: how to handle the explosion of the solution | October 15, Tuesday, 15:40 | METU, Institute of Applied Mathematics | Seminar Room

**October 15, 15:40, METU, IAM , Seminar Room**

In this talk, we first motivate the use of singular terminal condition : final trace of the solution of some semilinear PDEs or optimal control problem with constraint. Then we present some results concerning the existence of minimal supersolution for a wide class of generators. Finally we explain how the continuity problem can be solved in some specific cases. This talk is based on several papers with Thomas Kruse, Dmytro Marushkevych and Devin Sezer.

## Lecture 2: BSDE: extensions, an application in financial economics | October 16, Wednesday, 15:40 | Bilkent University, Department of Industrial Engineering | EA409

**Some extensions**

- Forward-backward SDEs
- Adding jumps or other kind of noises.
- Knightian uncertainty and second order BSDEs.

**The principal–agent problem**