Stochastic Processes 1000-135PS
1. Gaussian processes, stationary processes (1 lecture)
2. Poisson Process, generalized Poisson process (1 lecture)
3. Markov processes and operator semigroups, strong Markov property,
reflection principle, continuous time Markov chains. (5 lectures)
4. Diffusion processes and their relation to stochastic differential
equations (2 lectures)
5. Weak and strong solutions of stochastic differential equations.
Stroock-Varadhan Theorem on existence of a weak solution (sketch of the
proof), Yamada-Watanabe Theorem (with the proof) (4 lectures)
6. Processes with independent increments (Levy processes, stable
processes) (2 lectures)
Main fields of studies for MISMaP
Type of course
Course coordinators
Term 2023L: | Term 2024L: |
Learning outcomes
1. Knows the basic properties of the Wiener process and the Poisson process.
2. He knows the concept of the Markov process and is able to illustrate it with examples. He understands the concept of strong ownership
Markov and knows how to apply it.
3. Understands the basic relationships of Markov processes with the theory of semigroups.
4. Knows the concept of Markov chain with continuous time.
5. Understands the concept of the diffusion process, knows the Feynman-Kac formula and their relations with partial equations.
6. Can solve the Dirichlet problem using probabilistic methods.
Assessment criteria
Final grade will be based on students’ performance during the semester and final exam
Bibliography
1. I. Karatzas, S. Shreve, Brownian Motion and Stochastic Calculus. Springer-Verlag 1997.
2. D. Revuz, M. Yor, Continuous Martingales and Brownian Motion. Springer-Verlag 1999.
3. R. Schilling. L. Partzsch, Brownian Motion. An Introduction to Stochastic Processes. De Gruyter 2014.
4. A.D. Wentzell, Lectures on the theory of stochastic processes. PWN 1980
Additional information
Information on level of this course, year of study and semester when the course unit is delivered, types and amount of class hours - can be found in course structure diagrams of apropriate study programmes. This course is related to the following study programmes:
- Bachelor's degree, first cycle programme, Mathematics
- Master's degree, second cycle programme, Mathematics
Additional information (registration calendar, class conductors, localization and schedules of classes), might be available in the USOSweb system: