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.
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.