Outline

Overview

Capturing random phenomena is a challenging problem in many disciplines from biology, chemistry and physics through engineering to economics and finance. There is a wide spectrum of problems in these fields, which are described using random processes that evolve with time. Hence it is of crucial importance that applied mathematicians are equipped with tools used to analyse and quantify random phenomena. This unit will introduce an important class of stochastic processes, using the theory of martingales. You will study concepts such as the Ito stochastic integral with respect to a continuous martingale and related stochastic differential equations. Special attention will be given to the classical notion of the Brownian motion, which is the most celebrated and widely used example of a continuous martingale. By completing this unit, you will learn how to rigorously describe and tackle the evolution of random phenomena using continuous time stochastic processes. You will also gain a deep knowledge about stochastic integration, which is an indispensable tool to study problems arising, for example, in Financial Mathematics.

Unit details and rules

Academic unit	Mathematics and Statistics Academic Operations
Credit points	6
Prerequisites ?	None
Corequisites ?	None
Prohibitions ?	None
Assumed knowledge ?	A thorough knowledge of linear algebra (e.g., MATH2X22), stochastic processes (STAT3X21/4021) and probability theory (preferably STAT4528, or measure theoretic probability theory such as MATH3969/4069). Some familiarity with partial differential equations (e.g., MATH2X21 or MATH3X78/4078).
Available to study abroad and exchange students	Yes

Teaching staff

Coordinator	Marek Rutkowski, marek.rutkowski@sydney.edu.au

Assessment

The census date for this unit availability is 31 August 2026

Details
Criteria

Type	Description	Weight	Due	Length	Use of AI
Written exam	Final Exam Written exam for mathematical calculation/proof.	60%	Formal exam period	2 hours	AI prohibited
Outcomes assessed:
Written work	Assignment Written take-home assignment for mathematical calculation/proof.	15%	Week 05 Due date: 06 Sep 2026 at 23:59 Closing date: 16 Sep 2026	Submitted work	AI allowed
Outcomes assessed:
In-person written or creative task	Quiz Written in-class test for mathematical calculation/proof.	15%	Week 09 Due date: 07 Oct 2026 at 11:00 Closing date: 07 Oct 2026	1 hour	AI prohibited
Outcomes assessed:
Contribution	Contribution to tutorials Active contribution to solving mathematical problems during tutorials. Marking scheme: 2 marks for a correct solution, 1 mark for an attempted partial solution and no marks otherwise with the total up to 5 marks awarded during semester.	5%	Weekly	Weeks 1-13	AI allowed
Outcomes assessed:
Conversation	Attendance in tutorials Tutorial attendance and active participation in discussions. Marking scheme: 0.5 marks for each active attendance up to the total of 5 marks (with rounding up if required).	5%	Weekly	Weeks 1-13	AI allowed
Outcomes assessed:

Assessment summary

Quiz: Written in-class test for mathematical calculation/proof during lecture.

Assignment:听Written take-home assignment for mathematical calculation/proof.

Contribution:听Active contribution to solving mathematical problems during tutorials. Marking scheme: 2 marks for a correct solution, 1 mark for an attempted partial solution and no marks otherwise with the total up to 5 marks awarded during semester.

Attendance:听Tutorial attendance and active participation in discussions. Marking scheme: 0.5 marks for each active attendance up to the total of 5 marks (with rounding up if required).

Final Exam:听If a second replacement exam is required, this exam may be delivered via an alternative assessment method, such as viva voce (oral exam). The alternative assessment will meet the same learning outcomes as the original听exam. The format of the alternative assessment will be determined by the unit coordinator.

Detailed information for each assessment can be found on Canvas.

Assessment criteria

The University awards common result grades, set out in the听听(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听excellence.
Pass	50 - 64	Representing at least routine knowledge and understanding over a spectrum of topics and听important ideas and听concepts in the course.
Fail	0 - 49	When you don鈥檛 meet the learning outcomes of the unit to a satisfactory standard.

听
听

For more information see guide to grades.

Use of generative artificial intelligence (AI)

You can use generative AI tools for open assessments. Restrictions on AI use apply to secure, supervised assessments used to confirm if students have met specific learning outcomes.

Refer to the assessment table above to see if AI is allowed, for assessments in this unit and check Canvas for full instructions on assessment tasks and AI use.

If you use AI, you must always acknowledge it. Misusing AI may lead to a breach of the听Academic Integrity Policy.

Visit the听Current Students website听for more information on AI in assessments, including听details on how to acknowledge its use.

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.

This unit has an exception to the standard University policy or supplementary information has been provided by the unit coordinator. This information is displayed below:

IMPORTANT: Late penalties rules for this course are different and described below: 1) Deduction of 10% of the maximum mark for each calendar day after the due date. 2) After five calendar days late, a mark of zero will be awarded. 3) For all assessments, the rules for special consideration/arrangement apply. The maximal possible extension is 7 days.

Academic integrity

The University expects students to act ethically and honestly and will treat all allegations of academic integrity breaches seriously.

Our website听provides information on academic integrity and the resources available to all students. This includes advice on how to avoid common breaches of academic integrity. Ensure that you have completed the听Academic Honesty Education Module (AHEM)听which is mandatory for all commencing coursework students

Penalties for serious breaches can significantly impact your studies and your career after graduation. It is important that you speak with your unit coordinator if you need help with completing assessments.

Visit the听Current Students website听for more information on AI in assessments, including听details on how to acknowledge its use.

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听simple extension.鈥� 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鈥檛 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听special consideration or special arrangements.

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

Using AI responsibly

Co-created with students,听听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.

Support for students

The Support for Students Policy听reflects the University鈥檚 commitment to supporting students in their academic journey and making the University safe for students. It is important that you read and understand this policy so that you are familiar with the range of support services available to you and understand how to engage with them.

The University uses email as its primary source of communication with students who need support under the Support for Students Policy. Make sure you check your University email regularly and respond to any communications received from the University.

Learning resources and detailed information about weekly assessment and learning activities can be accessed via Canvas. It is essential that you visit your unit of study Canvas site to ensure you are up to date with all of your tasks.

If you are having difficulties completing your studies, or are feeling unsure about your progress, we are here to help. You can access the support services offered by the University at any time:

Support and Services (including health and wellbeing services, financial support and learning support)
Course planning and administration
Meet with an Academic Adviser

Weekly schedule

WK	Topic	Learning activity
Week 01	Conditional expectation.	Lecture (3 hr)
Week 02	Continuous-time stochastic processes.	Lecture (3 hr)
Week 02	Conditional expectation.	Tutorial (1 hr)
Week 03	Continuous-time martingales.	Lecture (3 hr)
Week 03	Continuous-time stochastic processes.	Tutorial (1 hr)
Week 04	Wiener Process and its sample paths.	Lecture (3 hr)
Week 04	Continuous-time martingales.	Tutorial (1 hr)
Week 05	Ito鈥檚 stochastic integral for continuous martingales.	Lecture (3 hr)
Week 05	Wiener Process and its sample paths.	Tutorial (1 hr)
Week 06	Ito鈥檚 formula for continuous semimartingales.	Lecture (3 hr)
Week 06	Ito鈥檚 stochastic integral for continuous martingales.	Tutorial (1 hr)
Week 07	Local time and Ito-Tanaka-Meyer formula.	Lecture (3 hr)
Week 07	Ito鈥檚 formula for continuous semimartingales.	Tutorial (1 hr)
Week 08	Stochastic exponential and Girsanov鈥檚 theorem.	Lecture (3 hr)
Week 08	Local time and Ito-Tanaka-Meyer formula.	Tutorial (1 hr)
Week 09	Predictable representation property of a Wiener process.	Lecture (3 hr)
Week 09	Stochastic exponential and Girsanov鈥檚 theorem.	Tutorial (1 hr)
Week 10	Stochastic differential equations.	Lecture (3 hr)
Week 10	Predictable representation property of a Wiener process.	Tutorial (1 hr)
Week 11	Feynman-Kac formula and PDEs.	Lecture (3 hr)
Week 11	Stochastic differential equations.	Tutorial (1 hr)
Week 12	Probability distributions for a Wiener process with drift.	Lecture (3 hr)
Week 12	Feynman-Kac formula and PDEs.	Tutorial (1 hr)
Week 13	Poisson process and its extensions.	Lecture (3 hr)
Week 13	Probability distributions for a Wiener process with drift.	Tutorial (1 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.

Outcomes
Graduate qualities

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

LO1. Demonstrate an understanding of the concepts of continuous time processes, stopping times and martingales.鈥�
LO2. 鈥婣pply the properties of Brownian motion (such as martingale property, Markov property, behaviour under the equivalent change of measure and predictable representation property). 鈥�
LO3. Describe the construction of Ito integral and apply Ito integration correctly in relevant examples.鈥�
LO4. Identify the similarities and differences between strong and weak solutions to SDEs and apply existence and uniqueness results to some specific SDEs.鈥�
LO5. Check the martingale property of stochastic processes using Ito's lemma.
LO6. Use the Feyman-Kac formula to analyse real world problems including arbitrage pricing.
LO7. Identify, formulate and solve original practical problems that can be addressed using mathematical techniques learnt in this unit and examine the implementations and provide an interpretation of the results.鈥�
LO8. Independently research sources provided by lecturers such as journal articles and working papers and evaluate the real-world plausibility and effectiveness of various approaches and tools.鈥�

Graduate qualities

The graduate qualities are the qualities and skills that all 爆料王 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鈥檚 personal and professional selves in an ethical context.
GQ9	Influence Engaging others in a process, idea or vision.

Outcome map

Learning outcomes	Graduate qualities
Learning outcomes

Responding to student feedback

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

More emphasis is put on regular work during the semester, students' participation and engagement is essential.

Disclaimer

Important: the 爆料王 regularly reviews units of study and reserves the right to change the units of study available annually. To stay up to date on available study options, including unit of study details and availability, refer to the relevant handbook.

To help you understand common terms that we use at the University, we offer an .

爆料王

Current students

Overview

Overview

Unit details and rules

Teaching staff

Assessment

Assessment

Assessment summary

Assessment criteria

Use of generative artificial intelligence (AI)

Late submission

Academic integrity

Learning support

Learning support

Simple extensions

Special consideration

Using AI responsibly

Support for students

Weekly schedule

Weekly schedule

Study commitment

Learning outcomes

Learning outcomes

Graduate qualities

Outcome map

Responding to student feedback

Responding to student feedback

Disclaimer

On this page

Useful links

爆料王

MATH4512: Stochastic Analysis

Semester 2, 2026 [Normal day] - Camperdown/Darlington, Sydney

Overview

Overview

Unit details and rules

Teaching staff

Assessment

Assessment

Assessment summary

Assessment criteria

Use of generative artificial intelligence (AI)

Late submission

Academic integrity

Learning support

Learning support

Simple extensions

Special consideration

Using AI responsibly

Support for students

Weekly schedule

Weekly schedule

Study commitment

Learning outcomes

Learning outcomes

Graduate qualities

Outcome map

Responding to student feedback

Responding to student feedback

Disclaimer

On this page

Useful links