Dr Luciano Campi COL.7.10

This course is available on the MSc in Risk and Stochastics, MSc in Statistics (Financial Statistics) and MSc in Statistics (Financial Statistics) (Research). This course is available with permission as an outside option to students on other programmes where regulations permit.

The course is offered as a regular examinable half-unit as well as a service to students and academic staff.

Analysis and algebra at the level of a BSc in pure or applied mathematics and basic statistics and probability theory with stochastic processes. Knowledge of measure theory is not required as the course gives a self-contained introduction to this branch of analysis.

The course covers core topics in measure theoretic probability and modern stochastic calculus, thus laying a rigorous foundation for studies in statistics, actuarial science, financial mathematics, economics, and other areas where uncertainty is essential and needs to be described with advanced probability models. Emphasis is on probability theory as such rather than on special models occurring in its applications. Brief review of basic probability concepts in a measure theoretic setting: probability spaces, random variables, expected value, conditional probability and expectation, independence, Borel-Cantelli lemmas Construction of probability spaces with emphasis on stochastic processes. Operator methods in probability: generating functions, moment generating functions, Laplace transforms, and characteristic functions. Notions of convergence: convergence in probability and weak laws of large numbers, convergence almost surely and strong laws of large numbers, convergence of probability measures and central limit theorems. If time permits and depending on the interest of the students topics from stochastic calculus might be covered as well.

20 hours of lectures and 10 hours of seminars in the MT.

Exercises are set weekly and solutions are discussed in the lectures. There will be two rounds of compulsory written coursework which will be marked, one in the MT and one in the LT.

Williams, D. (1991): Probability with Martingales. Cambridge University Press;

Kallenberg, O. (2002). Foundations of modern probability. Springer;

Billingsley, P. (2008). Probability and measure. John Wiley & Sons;

Jacod, J., & Protter, P. E. (2003). Probability essentials. Springer;

Dudley, R. M. (2002). Real analysis and probability (Vol. 74). Cambridge University Press

Exam (100%, duration: 2 hours) in the main exam period.