From 14.04.2014 to 19.07.2014

Monday: 14-16 MA-744 Mathematics building Strasse der 17 Juni

Wednesday: 10-12 MA-544 Mathematics building Strasse der 17 Juni

Backward Stochastic Differential Equations (BSDEs) is a relatively new topic in stochastics, but has been the subject of intense research ever since it was introduced twenty years ago. Indeed, beside the mathematical interest in this object, it has numerous applications in financial mathematics and constitutes in addition a bridge to partial differential equations or stochastic control theory. This lecture aims at covering both topics in classical BSDE theory and recent developments in the theory of minimal supersolutions, as well as the related applications to financial mathematics.

We put a particular emphasis on the different techniques used to derive existence and uniqueness of (super) solutions of BSDEs such as fixed points or compactness. We also study the structural properties of these solutions to derive some duality results. On the way, several central techniques in stochastic analysis will be covered: important inequalities, compactness theorems, (super)martingale results, Markov properties, dual spaces of processes, bounded mean oscillations martingales, among others. Throughout the whole course we illustrate the theory by means of examples within financial mathematics such as (super) hedging and pricing in incomplete markets.

The 4 weekly hours of the course correspond roughly to 3 hours of lecture and 1 hour of exercises. The lecture will be divided in two parts.

- BSDEs:
- existence and uniqueness of solutions of BSDEs (fixed point), Comparison Theorem (Lipschitz case)
- application in finance: pricing in incomplete markets
- relation to risk measures and non-linear expectations
- Markov-Property and relation to PDE
- how quadratic BSDEs arise within the theory of utility optimization

- Minimal Supersolutions of BSDEs:
- definition as a generalization of BSDEs
- existence and Uniqueness (compactness techniques)
- structural properties of the related nonlinear operator
- stability results
- duality
- Markov property

if time allows for it, we address the non dominated case and the notion of robustness and capacities.

The following methods and tools used – and most of the time proved in the lecture – covers

- martingale representation
- BMO martingales
- convex analysis, Fenchel-Moreau
- the spaces H p
- modifications, different kind of convergences
- supermartingales (convergence, Doob-Meyer)
- doubling strategies
- essinf
- Markov-property

WT I and II, FiMa I. Desirable WT III (stochastic analysis, can however be followed parallely) and FiMa II

[1] S. Drapeau, G. Heyne, and M. Kupper, Minimal supersolutions of convex bsdes, accepted for publication in Annals of Probability, (2011).

[2] N. El Karoui, S. Peng, and M. C. Quenez, Backward stochastic differential equations in finance, Mathematical Finance, 1 (1997), pp. 1–71.

[3] G. Heyne, M. Kupper, and C. Mainberger, Minimal supersolutions of BSDEs with lower semicontinuous generators, accepted for publication in Annales de l’Institut Henri Poincar ́e. Probabilit ́es et Statistiques, (2012).

[4] I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, vol. 113 of Graduate Texts in Mathematics, Springer-Verlag, New York, second ed., 1991.