Learning Outcomes:
On the completion of this module the student should be familiar with the fundamental concepts of Probability Theory which lead towards the deep mathematical theory of Stochastic Analysis. This includes independence, expectation, conditional expectation, stochastic processes, filtrations, martingales.
The student will develop their ability to deal with abstract concepts and to relate them to real world examples. The student's ability to realise and critique proofs and arguments will be enhanced.
Indicative Module Content:
Measure theoretic approach to probability theory; types of convergence for random sequences; independence of sigma algebras; Borel-Cantelli lemmas; laws of large numbers; conditional probability and expectation; martingale convergence theorems; optional stopping theorems for martingales. Time permitting, central limit theorems and/or Brownian motion.