Title

The mini course starts with a brief review of the stochastic integral process for continuous semimartingales, the covariation process, the integration-by-parts formula as well as Ito's isometry and Ito's formula, illustrated by several instructive examples. For the main results we present, we start with Levy's characterization of Brownian motion and measure changes (Girsanov's theorem). After an interlude on different versions of martingale convergence, we discuss the integral representation theorem for Brownian local martingales (needed for hedging financial derivatives), followed by Kazamaki's and Novikov's criterion for the uniform integrability of the stochastic exponential. This property allows to pass to an absolutely continuous or even equivalent probability measure. Time permitting, a short introduction to stochastic differential equations will be given. The emphasis of the presentation will be on the above results and illustrative examples; the detailed proofs are contained in accompanying lecture notes, which are available for participants of the course.