Unit of study_

# STAT4528: Probability and Martingale Theory

Probability Theory lays the theoretical foundations that underpin the models we use when analysing phenomena that involve chance. This unit introduces the students to modern probability theory (based on measure theory) that was developed by Andrey Kolmogorov. You will be introduced to the fundamental concept of a measure as a generalisation of the notion of length and Lebesgue integration which is a generalisation of the Riemann integral. This theory provides a powerful unifying structure that brings together both the theory of discrete random variables and the theory of continuous random variables that were introduced earlier in your studies. You will see how measure theory is used to put other important probabilistic ideas into a rigorous mathematical framework. These include various notions of convergence of random variables, 0-1 laws, conditional expectation, and the characteristic function. You will then synthesise all these concepts to establish the Central Limit Theorem and to thoroughly study discrete-time martingales. Originally used to model betting strategies, martingales are a powerful generalisation of random walks that allow us to prove fundamental results such as the Strong Law of Large Numbers or analyse problems such as the gambler's ruin. By doing this unit you will become familiar with many of the theoretical building blocks that are required for any in-depth study in probability, stochastic systems or financial mathematics.

Code STAT4528 Mathematics and Statistics Academic Operations 6
 Prerequisites: ? None None STAT4028 STAT2X11 or equivalent and STAT3X21 or equivalent; that is, a good foundational knowledge of probability and some acquaintance with stochastic processes

At the completion of this unit, you should be able to:

• LO1. Demonstrate a coherent and advanced knowledge of the concepts of measure theory and Lebesgue integration and how they provide a unified approach to a wide variety of problems arising in probability.
• LO2. Communicate mathematical analyses and solutions to mathematical and practical problems in probability and related fields clearly in a variety of media to diverse audiences.
• LO3. Apply characteristic function techniques to prove foundational theoretical results in probability.
• LO4. Compare and contrast different forms of stochastic convergence.
• LO5. Develop a theoretical tool set using martingales and Brownian motion for constructing solutions to a broad suite of problems in statistics, mathematical finance and other applied fields.
• LO6. Devise solutions to novel mathematical problems in probability theory.
• LO7. Understand the concept of martingale and its basic properties, and be able to recognise important examples of martingales arising Statistics, Finance and Probability theory
• LO8. Be able to use the Optional Stopping Theorem in order to compute some important probabilities and expectations and understand its limitations

## Unit outlines

Unit outlines will be available 2 weeks before the first day of teaching for the relevant session.