Introductory description

This module runs is available for students on a course where it is a listed option and as an Unusual Option to students who have completed one of the prerequisite modules.

Pre-requisite(s): MA359 Measure Theory, or ST342 Mathematics of Random Events/ST3xx Measure Theory for Probability..

Leads to: ST401 Stochastic Methods in Finance, ST402 Risk Theory, ST403 Brownian Motion, ST411 Dynamic Stochastic Control

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

• Independence and zero-one laws
• Modes of convergence for sequences of random variables
• Limit theorems: law of large numbers (LLN) and central limit theorems (CLT)
• Conditioning and discrete time martingales

Learning outcomes

By the end of the module, students should be able to:

• understand the different modes of convergence for sequences of random and the relationship between these different modes;
• state and prove the Central Limit Theorem via the method of characteristic functions and understand how this result can be applied;
• understand some basic results on discrete -time martingales, including the martingale convergence theorem and optional stopping theorem, and show how these results can be used to obtain various characteristics of simple random walks.
• understand the ideas relating to independence and zero-one laws and be able to apply these ideas in simple contexts;

Indicative reading list

View reading list on Talis Aspire

Subject specific skills

• Demonstrate facility with rigorous probabilistic methods.

• Evaluate, select and apply appropriate mathematical and/or probabilist techniques.

• Demonstrate knowledge of and facility with formal probability concepts, both explicitly and by applying them to the solution of mathematical problems.

• Create structured and coherent arguments communicating them in written form. 

• Construct logical mathematical arguments with clear identification of assumptions and conclusions.

• Reason critically, carefully, and logically and derive (prove) mathematical results.

Transferable skills

• Problem solving: Use rational and logical reasoning to deduce appropriate and well-reasoned conclusions. Retain an open mind, optimistic of finding solutions, thinking laterally and creatively to look beyond the obvious. Know how to learn from failure.

• Self awareness: Reflect on learning, seeking feedback on and evaluating personal practices, strengths and opportunities for personal growth.

• Communication: Present arguments, knowledge and ideas, in a range of formats.

• Professionalism: Prepared to operate autonomously. Aware of how to be efficient and resilient. Manage priorities and time. Self-motivated, setting and achieving goals, prioritising tasks.