Unit outline_

# STAT4528: Probability and Martingale Theory

## Overview

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.

### Unit details and rules

Academic unit Mathematics and Statistics Academic Operations 6 None None STAT4028 STAT2X11 or equivalent and STAT3X21 or equivalent; that is, a good foundational knowledge of probability and some acquaintance with stochastic processes Yes

### Teaching staff

Coordinator Ben Goldys, beniamin.goldys@sydney.edu.au Ben Goldys

## Assessment

Type Description Weight Due Length
Supervised exam

Final exam
Written examination
60% Formal exam period 2 hours
Outcomes assessed:
Assignment Assignment 1
For STAT4528 only
10% Week 04
Due date: 19 Mar 2023 at 23:59

Closing date: 19 Mar 2023
10 days
Outcomes assessed:
Assignment Assignment 2
For STAT4528 only
10% Week 07
Due date: 09 Apr 2023 at 23:59

Closing date: 09 Apr 2023
10 days
Outcomes assessed:
Assignment Assignment 3
Submitted solutions to extended problems.
20% Week 12
Due date: 19 May 2023 at 23:59

Closing date: 19 May 2023
2 weeks
Outcomes assessed:

### Assessment summary

• Assignment 1: will contain 6 questions testing the understanding of the first part of the course focused on probabilistic measure theory.
• Assignment 2: will consist of 4 questions testing the understanding of the theory of martingales and their applications in probability and statistics.
• Final exam: The exam will cover all material in the unit from both lectures and tutorials. The exam will have questions to answer. It is a 2h exam during the formal exam period.
• Final exam: If a second replacement exam is required, this exam may be delivered via an alternative assessment method, such as a viva voce (oral exam). The alternative assessment will meet the same learning outcomes as the original exam. The format of the alternative assessment will be viva voce and details will be provided later.

Detailed information for each assessment would be given during the tutorial or can be found on Canvas

### Assessment criteria

The University awards common result grades, set out in the Coursework Policy 2014 (Schedule 1).

As a general guide, a high distinction indicates work of an exceptional standard, a distinction a very high standard, a credit a good standard, and a pass an acceptable standard.

Result name

Mark range

Description

High distinction

85 - 100

Representing complete or close to complete mastery of the material.

Distinction

75 - 84

Representing excellence, but substantially less than complete mastery.

Credit

65 - 74

Representing a creditable performance that goes beyond routine knowledge and understanding, but less than excellence.

Pass

50 - 64

Representing at least routine knowledge and understanding over a spectrum of topics and important ideas and concepts in the course.

Fail

0 - 49

When you don’t meet the learning outcomes of the unit to a satisfactory standard.

### Late submission

In accordance with University policy, these penalties apply when written work is submitted after 11:59pm on the due date:

• Deduction of 5% of the maximum mark for each calendar day after the due date.
• After ten calendar days late, a mark of zero will be awarded.

The Current Student website  provides information on academic integrity and the resources available to all students. The University expects students and staff to act ethically and honestly and will treat all allegations of academic integrity breaches seriously.

We use similarity detection software to detect potential instances of plagiarism or other forms of academic integrity breach. If such matches indicate evidence of plagiarism or other forms of academic integrity breaches, your teacher is required to report your work for further investigation.

You may only use artificial intelligence and writing assistance tools in assessment tasks if you are permitted to by your unit coordinator, and if you do use them, you must also acknowledge this in your work, either in a footnote or an acknowledgement section.

Studiosity is permitted for postgraduate units unless otherwise indicated by the unit coordinator. The use of this service must be acknowledged in your submission.

## Learning support

### Simple extensions

If you encounter a problem submitting your work on time, you may be able to apply for an extension of five calendar days through a simple extension.  The application process will be different depending on the type of assessment and extensions cannot be granted for some assessment types like exams.

### Special consideration

If exceptional circumstances mean you can’t complete an assessment, you need consideration for a longer period of time, or if you have essential commitments which impact your performance in an assessment, you may be eligible for special consideration or special arrangements.

Special consideration applications will not be affected by a simple extension application.

### Using AI responsibly

Co-created with students, AI in Education includes lots of helpful examples of how students use generative AI tools to support their learning. It explains how generative AI works, the different tools available and how to use them responsibly and productively.

## Weekly schedule

WK Topic Learning activity Learning outcomes
Week 01 Introduction to Measure Theory Lecture and tutorial (4 hr)
Week 02 Sigma Algebras and measurable functions Lecture and tutorial (4 hr)
Week 03 Lebesgue Integrals Lecture and tutorial (4 hr)
Week 04 Convergence Lecture and tutorial (4 hr)
Week 05 Product Measures Lecture and tutorial (4 hr)
Week 06 Central Limit Theorem Lecture and tutorial (4 hr)
Week 07 Conditional Expectation Lecture and tutorial (4 hr)
Week 08 Discrete Time Martingales Lecture and tutorial (4 hr)
Week 09 Discrete Stopping Times Lecture and tutorial (4 hr)
Week 10 Continuous Time Martingales Lecture and tutorial (4 hr)
Week 11 Continuous Stopping Times and Local Martingales Lecture and tutorial (4 hr)
Week 12 Brownian motion; semimartingales Lecture and tutorial (4 hr)
Week 13 Revision week Lecture and tutorial (4 hr)

### Study commitment

Typically, there is a minimum expectation of 1.5-2 hours of student effort per week per credit point for units of study offered over a full semester. For a 6 credit point unit, this equates to roughly 120-150 hours of student effort in total.

## Learning outcomes

Learning outcomes are what students know, understand and are able to do on completion of a unit of study. They are aligned with the University's graduate qualities and are assessed as part of the curriculum.

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

The graduate qualities are the qualities and skills that all University of Sydney graduates must demonstrate on successful completion of an award course. As a future Sydney graduate, the set of qualities have been designed to equip you for the contemporary world.

 GQ1 Depth of disciplinary expertise Deep disciplinary expertise is the ability to integrate and rigorously apply knowledge, understanding and skills of a recognised discipline defined by scholarly activity, as well as familiarity with evolving practice of the discipline. GQ2 Critical thinking and problem solving Critical thinking and problem solving are the questioning of ideas, evidence and assumptions in order to propose and evaluate hypotheses or alternative arguments before formulating a conclusion or a solution to an identified problem. GQ3 Oral and written communication Effective communication, in both oral and written form, is the clear exchange of meaning in a manner that is appropriate to audience and context. GQ4 Information and digital literacy Information and digital literacy is the ability to locate, interpret, evaluate, manage, adapt, integrate, create and convey information using appropriate resources, tools and strategies. GQ5 Inventiveness Generating novel ideas and solutions. GQ6 Cultural competence Cultural Competence is the ability to actively, ethically, respectfully, and successfully engage across and between cultures. In the Australian context, this includes and celebrates Aboriginal and Torres Strait Islander cultures, knowledge systems, and a mature understanding of contemporary issues. GQ7 Interdisciplinary effectiveness Interdisciplinary effectiveness is the integration and synthesis of multiple viewpoints and practices, working effectively across disciplinary boundaries. GQ8 Integrated professional, ethical, and personal identity An integrated professional, ethical and personal identity is understanding the interaction between one’s personal and professional selves in an ethical context. GQ9 Influence Engaging others in a process, idea or vision.

### Outcome map

GQ1 GQ2 GQ3 GQ4 GQ5 GQ6 GQ7 GQ8 GQ9

## Responding to student feedback

This section outlines changes made to this unit following staff and student reviews.

No changes have been made since this unit was last offered.