Entry requirements

120 credits including 90 credits in Mathematics. Integration Theory.

Learning outcomes

In order to pass the course the student should be able to

• construct probability measures on finite dimensional Euclidean spaces;
• use measure-theoretic and analytical techniques for the proof of limit theorems in probability;
• effectively use conditional expectations and Radon-Nikodym derivatives;
• outline proofs of important theorems of the course, and explain the main ideas of the proofs;
• characterise random variables with values in Euclidean and function spaces;
• use symmetry and invariance arguments;
• construct and compute with normal random variables in Euclidean spaces by cogently using basic notions of linear algebra;
• give examples of applications of probability, construct simple stochastic models and solve stochastic equations;
• decelop basic understanding of the fundamental random variables of probability theory, such as Brownian motion and the Poisson process.

Content

Quick review of basic probability. Probability spaces. Coin tosses and measure-theoretic foundations. Integration. Inequalities. Radon-Nikodym theorem and conditional probability. Convergence. Fundamental theorems of probability. Martingales. Characteristic functions. Gaussian spaces. Brownian motion. Point processes and Poisson. Other classes of function-valued random variables. Applications and additional topics.

Instruction

Lectures and problem solving sessions.

Assessment

The assessment will be based on individual projects at the end of the course. Assignments during the course.