Instructions for exam NMSA405
The exam is oral. All material covered during the lecture
could be part of the exam.
First the student will draw one of the following
8 topics and have some time to prepare the answer (it could be in English,
in Czech or combination of both).
Afterwards some additional questions
(from any topic covered during the semester) could be asked.
Topics
1. random sequence, product σ-algebra, random element with values in the space of real sequences,
distribution uniquelly determined by finite-dimensional distributions, Daniell's extension theorem
2. stopping time, filtration, stopping time σ-algebra, examples, properties and relations,
strong Markov property of a random walk
3. (sub)martingale, definition and examples, stability of (sub)martingale property with respect to filtration
and convex transformation, Doob decomposition theorem
4. optional stopping theorem, optional sampling theorem, Wald's equations
5. bankruptcy is definitive, Doob's maximal inequalities, Kolmogorov inequality
6. Doob's upcrossing inequality, Doob's submartingale convergence theorem
7. convergence of uniformly integrable (sub)martingale, continuity of conditional expectation in condition,
submartingale converges or explodes
8. limit theorems for martingale differences
Grading
1 (excellent): student knows the proofs, clearly understands the material and is able to use it
2 (very good): slight flaws
3 (good): student knows only simple proofs or has problems with explanation of theoretical results and their
application
4 (failed): student is unable to correctly formulate some definition or theorem (even if other
answers are satisfactory) or shows lack of understanding of the material
Dates
15.1., 20.1., 27.1., 2.2., 10.2., 17.2., 29.5.