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

• introduce the stochastic exponential and the stochastic logarithm, explain basic properties and characterisations, 
• use Lévy's characterization of Brownian motion, 
• formulate Girsanov's theorem and apply it to adjust the drift of Brownian motion by a measure change, 
• explain Doob's upcrossing inequality and derive Doob's convergence theorems for submartingales,
• explain and apply the predictable integral representation theorem for Brownian local martingales, 
• check and derive conclusions from Kazamaki's and Novikov's criterion,
• explain the different notions of a solution of a stochastic differential equation (SDE) and illustrate the basic effects by examples,
• derive the solution in the linear case and discuss its uniqueness,
• define the Ornstein–Uhlenbeck process and derive its basic properties,
• apply the extended Grönwall inequality and illustrate the necessity of some of its assumptions by corresponding counterexamples,
• investigate SDEs for existence and uniqueness under Lipschitz and boundedness conditions, and estimate the moments of the solution,
• describe and apply the ideas and methods used to prove tha main theorems of the course.

Stochastic exponential of continuous semimartingales, stochastic logarithm, Lévy's characterization of standard Brownian motion, Girsanov's theorem, change of drift using Girsanov's theorem, Doob's upcrossing inequality, Doob's convergence theorems for submartingales, representation of Brownian local martingales, Kazamaki's and Novikov's criterion, stochastic differential equations (examples, terminology, solution in the linear case), Ornstein–Uhlenbeck process, extended Grönwall inequality, existence and uniqueness of strong solutions under Lipschitz and boundedness conditions, moment estimates.

The basic contents and concepts are presented by the head of the LVA and illustrated and discussed with the help of examples.

The performance is assessed by an oral examination at the end of the semester.

Registered students (to part 1 of the course) 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.