Definition and properties of multi-dimensional normal distribution, definition and elementary properties of Brownian motion, existence and Hölder continuity of Brownian motion using the Kolmogorov-Chentsov continuity criterion, filtrations, stopping times, progressive measurability, path properties, martingales, uniform integrability, Vitali's convergence theorem, sub- and supermartingales, maximum inequality, Doob's inequality for p-integrable submartingales, Doob's optional sampling theorem with applications, local martingales and examples, integration of predictable step processes, p-variation of functions, quadratic variation and covariation process of continuous local martingales, Kunita-Watanabe inequality, stochastic integration for continuous local martingales and generalization for continuous semimartingales
Registered students have access to an English script in electronic format with numerous references. The script will be updated on a continuing basis.
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.
Ioannis Karatzas und Steven E. Shreve: Brownian Motion and Stochastic Calculus. 2. Edition, Springer-Verlag, ISBN 0-38797-655-8.
Bernt Øksendal: Stochastic Differential Equations: An Introduction with Applications. 6. Edition, Springer-Verlag, 2007, ISBN 978-3-54004-758-2.
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.
