Syllabus
1. Conditional expectation w.r.t. sigma-algebra, random process, finite-dimensional distributions, Daniell-Kolmogorov and Kolmogorov-Chentsov theorems.
2. Martingales, definition of super- and submartingales, filtration, basic examples. Stopping times and hitting times of a subset of the state space by a random process. Maximal inequalities, Doob-Meyer decomposition.
3. Quadratic variation of martingales, Wiener process and its basic properties.
4. Stochastic integration w.r.t. Wiener process, definition and basic properties. Stochastic differential and Ito formula, examples.
5. Stochastic integration w.r.t. martingales - an introduction.
Annotation
The main objective is to introduce the fundamentals of probability theory that are used in finance and insurance mathematics. The central concepts here are conditional expectation and discrete and continuous martingales that will be introduced and explained. Their basic properties will be studied and the most important examples (Wiener process and stochastic integral) will be examined. Basics of the stochastic calculus will be introduced and studied
(Ito Lemma). These techniques form the fundamentals for investigation of stochastic models in finance and insurance mathematics.