Dr Pavel Gapeev

This course is available on the MSc in Applicable Mathematics, MSc in Financial Mathematics and MSc in Risk and Stochastics. This course is available as an outside option to students on other programmes where regulations permit.

Some background in real analysis is essential.

The purposes of this course are (a) to explain the formal basis of abstract probability theory, and the justification for basic results in the theory, and (b) to explore those aspects of the theory most used in advanced analytical models in economics and finance. The approach taken will be formal. Probability spaces and probability measures. Random variables. Expectation and integration. Convergence of random variables. Conditional expectation. The Radon-Nikodym Theorem. Martingales. Stochastic processes. Brownian motion. The Itô integral.

20 hours of lectures and 10 hours of seminars in the MT. 1 hour of seminars in the LT.

Full lecture notes will be provided. The following may prove useful: J S Rosenthal, A First Look at Rigorous Probability Theory; G R Grimmett & D R Stirzaker, Probability and Random Processes; D Williams, Probability with Martingales; M Caplinski & E Kopp, Measure, Integral and Probability; J Jacod & P Protter, Probability Essentials.

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