Introduction to stochastic analysis as needed for continuous-time financial and actuarial mathematics.

Review of basic definitions from probability theory, Kolmogorov's extension theorem, definition and basic properties of the multi-dimensional normal distribution, Brownian motion/Wiener process, Gaussian processes, Kolmogorov-Chentsov criterion for continuous sample paths, existence proof of Brownian motion, definition of the Itô integral, Itô isometry, martingales and martingale inequalities, basic properties of the Itô integral, existence of continuous versions, one- and multi-dimensional Itô formula, applications, martingale representation

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.