After successful completion of the course, students are able to...

• explain and use the definition and properties of multidimensional normal distribution and related distributions,
• list the definition and elementary properties of Brownian motion and to sketch the proof of its existence and Hölder continuity by means of the of Kolmogorov-Chentsov continuity criterion,
• put filtrations, stopping times, progressive measurability and path properties of processes in relation to each other,
• explain martingales, sub- and supermartingales, uniform integrability and Vitali's convergence theorem,
• apply Doob's classical results (maximum inequalities, L^p inequality, optional stopping theorem) and sketch their proofs,
• discuss local martingales and to give examples for strict local martingales
• integrate predictable step processes, introduce the quadratic variation and the covariation process for continuous local martingales and calculate these processes for some examples,
• derive the existence of the stochastic integral for continuous local martingales by means of the Kunita-Watanabe inequality and explain the generalization to continuous semimartingales,
• explain and apply the chain rule, integration by parts, and convergence theorems for stochastic integrals (w.r.t. continuous semimartingales),.

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, chain rule and convergence theorems for stochastic integrals (with respect to continuous semimartingales), integration by parts, multi-dimensional Ito formula with applications, Tanaka's formula, local Ito formula and Ito formula for holomorphic functions,

Presentation and derivation of the results by the lecturer on the blackboard, self-study of the lecture notes. Active participation in the accompanying exercises is strongly recommended; numerous exercises are included in the lecture notes.

Oral examination

Registered students have access to an English script in electronic format with numerous references. The script will be updated on a continuing basis. It contains study assignments. 

Additional literature:
Olav Kallenberg: Foundations of Modern Probability. 3. Edition, Springer-Verlag, 2021, ISBN 978-3-030-61871-1.
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.