Stochastic analysis as needed for continuous-time financial and actuarial mathematics.

Gronwall's inequality, examples and solution methods for stochastic differential equations, existence and uniqueness theorem, weak and strong solutions, Bayes' formula, Lévy's characterization of Brownian motion, several versions of Girsanov's theorem, exponential martingales, Kazamaki's criterion, Novikov's condition, Itô diffusions, Markov property, strong Markov property, generator of an Itô diffusion, Dynkin's formula, applications, characteristic operator, filtering problem

oral exam

Bernt Øksendal: Stochastic Differential Equations: An Introduction with Applications. 6. Edition, Springer-Verlag, 2007, ISBN 978-3-54004-758-2. Additional literature: Olav Kallenberg: Foundations of Modern Probability. 2. Edition, Springer-Verlag, 2002, ISBN 0-387-953113-2. Daniel Revuz and Marc Yor: Continuous Martingales and Brownian Motion, 3. Edition, Springer-Verlag, 1999, ISBN 3-540-64325-7. Foundations: David Williams: Probability with Martingales. Cambridge University Press, 1991,ISBN 0-521-40605-6. Heinz Bauer: Maß- und Integrationstheorie. 2. Edition, De Gruyter, 1992, ISBN 3-11013-626-0. Heinz Bauer: Wahrscheinlichkeitstheorie. 5. Edition, De Gruyter, 2002, ISBN 3-11017-236-4.