Review of basic definitions from probability theory, continuity theorem of Levy, definition and basic properties of the multi-dimensional normal distribution, Brownian motion/Wiener process, Gaussian processes, existence proof of Brownian motion, definition of the Itô integral, Itô isometry, martingales and martingale inequalities, basic properties of the Itô integral, one- and multi-dimensional Itô formula, martingale representation
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 und Marc Yor: Continuous Martingales and Brownian Motion, 3. Edition, Springer-Verlag, 1999, ISBN 3-540-64325-7.
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.
