This is the first announcement about the plan for the course Probability II. The goal is to study (discrete) stochastic process, that is a sequence of random variables defined on a common probability space.
For concreteness, I will focus on real-valued random variables and I plan to discuss several applications during the lectures and exercise sessions.
There will be three main topics:
1- Conditional expectation
2- Theory of martingales
3- Markov chains
These concepts are fundamental both in theoretical and applied probability. In particular, they have crucial applications in mathematical finance, data science and mathematical physics.
From a mathematical viewpoint, this course relies on analysis, especially measure theory, but also has connections to linear algebra and combinatorics.
In the first week, I will begin this course by a review of Probability I.
In the first lecture, this includes a short introduction to the foundations of probability theory, such as the key concepts of Lebesgue measure, random variables, probability law, independence, etc. I also plan to review the main modes of convergence for sequences of random variables.
In the second lecture, I will review the main results from Probability I: the Borel-Cantelli Lemma, the law of large numbers and central limit Theorems. I also plan to recall the concept of characteristic functions.
Information about the exercise classes Exercise classes (and lectures) will be streamed live and recorded. Submission of weekly assignment sheets is optional. There is no required minimum number of points to access the final exam. The first Assignment Sheet will be uploaded at the latest by Monday 21.09.2020.