Martingale representation, Gronwall's inequality, examples and solution methods for stochastic differential equations, existence and uniqueness theorem, Yamada-Watanabe condition, weak and strong solutions, Bayes' formula, Lévy's characterization of Brownian motion, several versions of Girsanov's theorem, exponential martingales, Kazamaki's criterion, Novikov's condition, Doob-Meyer decomposition, Itô diffusions, Markov property, strong Markov property, generator of an Itô diffusion, Dynkin's formula, applications
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.
Ioannis Karatzas und Steven E. Shreve: Brownian Motion and Stochastic Calculus. 2. Edition, Springer-Verlag, ISBN 0-38797-655-8.
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.
