The main goal of this course is to provide an introduction to the theory of Backward Stochastic Di fferential Equations (BSDEs) and to its strong connection to Finance. We will survey the classical existence/uniqueness results for this equations together with some proofs making the course accessible to anyone who is not familiar with the theory of BSDEs, and we will present the link between BSDEs and several problems in Finance, like for instance the utility maximization problem, superheding and quantitle hedging problems. For each of these questions, we will review the by now, well-established results and present some new developments. The course will be divided into fours lectures and we provide below a brief description of them.

**Lecture 1:** In the first lecture, we will rst show how BSDEs naturally arise in the study of several problems in Finance, like the utility maximization problem or for representing risk measures. Then, we will provide general existence/uniqueness results for Lipschitz and quadratic BSDEs.

**Lecture 2:** In the second lecture, we will see that in order to go further in the description of the solution to the utility maximization problem, one needs to consider a generalization of BSDEs namely Forward-Backward SDEs (FBSDEs). We will present the derivation of this system of equations and some theoretical results that can be derived for these equations.

**Lecture 3:** In this lecture, we will consider a slightly di fferent class of problems in Finance, which are related to the superhedging problem and which are known under the name of "quantile hedging" or "Stochastic Target problems with controlled loss". We will once again transpose this practical problem into the study of a new class of equations called BSDEs with weak terminal conditions.

**Lecture 4:** Finally, we will deal with a theoretical issue which is motivated by the numerical analysis of BSDEs, that is, the existence of densities for the marginal laws of the solution to a BSDE. This issue has been very few studied and we will revisit and extend results of the literature. Our approach is based on the use of the Malliavin calculus and of the Nourdin-Viens' formula that will be briefy introduced during this lecture.

**Locations:**

22.04.2014, 3 p.m. - 5 p.m., HU Berlin, Rudower Chaussee 25, Room 1.023 (BMS Lounge)

23.04.2014, 3 p.m. - 5 p.m., WIAS Berlin, Mohrenstraße 39, Room 406

24.04.2014, 2 p.m. - 4 p.m., TU Berlin, Straße des 17. Juni 136, Room MA748 (RTG Lounge)

25.04.2014, 2 p.m. - 4 p.m., TU Berlin, Straße des 17. Juni 136, Room MA748 (RTG Lounge)