Stochastic Processes (7.5hp)

This is a 7.5HP course for Ph.D. students in Statistics. It is planned as a traditional on campus blackboard lecture and exercise session course. The schedule is to be decided yet, but tentatively it will be blocked into two intensive ca week long blocks. Please email me directly, krzysztof_dot_bartoszek_at_liu_dot_se if you are interested in participating.

Purpose : This course gives a solid background in and understanding of important results and methods in stochastic processes at an advanced level. The objective of the course is to give students a solid knowledge of the construction of stochastic processes and fundamental concepts and theorems required for their analysis. The course will equip the student with knowledge supporting the modelling of the dynamics of random phenomena. Students should be able to explain the classification of states, verification of the martingale property, finding a stationary distribution, various convergence concepts in probability.

Contents:
1. Revision of selected parts of probability theory, in particular the moment generating function.
2. Stochastic processes - definition and examples.
3. Finite dimensional distributions of a stochastic process.
4. Homogeneous and non-homogeneou Poisson processes.
5. Markov chains, random walks, stochastic matrices.
6. Branching processes.
7. Martingales.
8. Doob Theorem.
9. Gaussian processes, Brownian motion.
10. Kolmogorov Theorem.

Examination and grades :
Course assessment consists of: oral presentation and/or written assignment. The grades given are pass or fail.

Literature :
G. Grimmett, D., Stirzaker, Probability and Random Processes, Oxford University Press, 2020.
S. Ross, Stochastic Processes, John Wiley and Sons, 1996.