This course gives a rigorous mathematical introduction to stochastic processes, stochastic differential equations, and their applications in finance. The first half of the course covers martingales, Poisson processes, Brownian motion, Ito integration, and stochastic differential equations driven by a Brownian motion. The second half of the course covers a range of more advanced topics to be selected by the students. This includes option pricing and investment optimisation, basics of Malliavin's stochastic calculus of variations, Black-Scholes formula and hedging, as well as aspects of the relationship between stochastic analysis and partial differential equations. 

Stochastic Analysis with Financial Applications provides an accessible but mathematically rigorous introduction to financial mathematics and quantitative finance.

This course is co-taught with undergraduate students but assessed separately. Assessment will include a research project.

Learning Outcomes

Upon successful completion, students will have the knowledge and skills to:

1. Demonstrate a deep understanding of the core mathematical tools and fundamental concepts of modern financial mathematics;
2. Use stochastic calculus efficiently in mathematical and financial problems, including option pricing;
3. Demonstrate strong capabilities for advanced mathematical reasoning, analysis and modeling linked to the theory of stochastic processes.
4. Use deep knowledge and understanding of stochastic analysis formulate responses to complex problems in finance.

Research-Led Teaching

The first part of the course is a classic undergraduate course, emphasising depth of understanding, and going through the construction of Brownian motion and Ito integration, and through the resolution of stochastic differential equations, in full detail, before applying the theory to option pricing. The second part of the course, however, operates as a transdiciplinary research seminar, emphasising breadth. It covers a large amount of advanced material in mathematics, finance (and potentially other disciplines) through short presentations. Discussions of this material include a range of perspectives, from academia and industry, and from various academic disciplines, while highlighting the specific voice of mathematicians.

Recommended Resources

“Stochastic Calculus and Financial Applications” by J. Michael Steele

There are a variety of online platforms you will use to participate in your study program at ANU, across all of your courses. These could include videos for lectures and other instruction, two-way video conferencing for interactive learning, email and other messaging tools for communication, interactive web apps for formative and collaborative activities, print and/or photo/scan for handwritten work and drawings, and home-based assessment.

ANU outlines recommended student system requirements to ensure you are able to participate fully in your learning. Other information is also available about the various Learning Platforms you may use.

Staff Feedback

Students are given feedback in this course in a variety of ways, matching the variety of assessment methods. This includes:

1. Oral feedback from the lecturer to the whole class regarding assignments and exams.
2. Written and oral feedback from the demonstrator on assignments.
3. Oral personal feedback from the lecturer through appointments.
4. Direct feedback on oral presentations just after the student’s talks.
5. Written feedback by the lecturer on the essay.

Student Feedback

ANU is committed to the demonstration of educational excellence and regularly seeks feedback from students. Students are encouraged to offer feedback directly to their Course Convener or through their College and Course representatives (if applicable). The feedback given in these surveys is anonymous and provides the Colleges, University Education Committee and Academic Board with opportunities to recognise excellent teaching, and opportunities for improvement. The Surveys and Evaluation website provides more information on student surveys at ANU and reports on the feedback provided on ANU courses.

Other Information

Use of Artificial Intelligence (AI): invigilated assessment tasks prohibit the use of electronic devices, and no AI usage is allowed in such tasks. For other assessment tasks, the use of generative AI will be governed by the class policy as specified on Canvas or as part of the individual assessment task instructions.

Class Schedule

Week/Session	Summary of Activities	Assessment
1	Conditional expectation; Worksheet 1
2	Discrete time martingales; Worksheet 1
3	Martingales estimates; Worksheet 2
4	Brownian motion; Worksheet 3
5	Properties of Brownian motion; Worksheet 4
6	Ito's integral; Worksheet 5	Assignment 1 due
7	Ito's formula; Worksheet 6	Assignment 2 due
8	Stochastic differential equations; Worksheet 7
9	Mathematical finance; Worksheet 8
10	Practice and exam preparation	Exam on Monday 12 October 2pm to 5pm
11	Seminar phase	Discussion of oral presentation videos
12	Seminar phase	Discussion of oral presentation videos; final essay due the Friday at start of exam period

Tutorial Registration

Workshop timing will be as allocated on MyTimetable. ANU utilises MyTimetable to enable students to view the timetable for their enrolled courses, browse, then self-allocate to small teaching activities / tutorials so they can better plan their time. Find out more on the Timetable webpage.

Assessment Summary

Assessment task	Value	Due Date	Return of assessment	Learning Outcomes
Two Assignments	25 %	*	*	1,2,3
Exam	25 %	12/10/2026	19/10/2026	1,2,3
Oral presentation videos and discussion	25 %	*	*	1,2,3,4
Essay	25 %	06/11/2026	13/11/2026	1,2,3,4

* If the Due Date and Return of Assessment date are blank, see the Assessment Tab for specific Assessment Task details

Policies

ANU has educational policies, procedures and guidelines, which are designed to ensure that staff and students are aware of the University’s academic standards, and implement them. Students are expected to have read the Academic Misconduct Rule before the commencement of their course. Other key policies and guidelines include:

• Student Assessment (Coursework) Policy and Procedure
• Special Assessment Consideration Policy and General Information
• Student Surveys and Evaluations
• Deferred Examinations
• Student Complaint Resolution Policy and Procedure

Assessment Requirements

The ANU is using Turnitin to enhance student citation and referencing techniques, and to assess assignment submissions as a component of the University's approach to managing Academic Integrity. For additional information regarding Turnitin please visit the ANU Online website Students may choose not to submit assessment items through Turnitin. In this instance you will be required to submit, alongside the assessment item itself, hard copies of all references included in the assessment item.

Moderation of Assessment

Marks that are allocated during Semester are to be considered provisional until formalised by the College examiners meeting at the end of each Semester. If appropriate, some moderation of marks might be applied prior to final results being released.

Participation

The course operates as a flipped classroom. This means that students first learn the content before meeting with the lecturer, and then use the class time to reach a deep level of understanding. Please see the course Canvas page for more information.

Examination(s)

This course includes a 25% exam, scheduled for Monday 12 October from 2pm to 5pm. This is the assessment activity appearing in MyTimetable.

Assessment Task 1

Two Assignments

There are two assignments, due on Fridays at the end of Weeks 6 and 7 respectively. It is expected that assignments will be returned one week after submission.

Task: prove several statements similar to the statements discussed in lectures and workshops. Hand in a complete typed or scanned personal solution.

Assignments are formative assessments. They are opportunities for you to develop your understanding of the material. To make the most of this opportunity you should do the following.

1. Attempt the questions yourself, as if you were in an exam situation. 
2. Look for knowledge you may be missing in the lecture notes, textbooks, or using online ressources (including AI). You want to be fully aware of what your current level of dependency on such resources is, and aim to decrease it.
3. Discuss the questions with your peers and/or lecturer (including in workshops).
4. Write up your answers as an individual writing exercise. 
5. Potentially check your answers, through discussions with peers and/or AIs.
6. Finalise your answers, and reflect on how much support you have used to generate them. This reflection needs to be recorded on your assignment. (Please see Canvas for more information about reflection statements.)

In comparison to MATH3015 assignment submissions, more maturity in mathematical writing is expected of MATH6115 students. The proofs don’t just need to be correct; they need to be pleasant to read.

Assessment Task 2

Exam

This is a traditional closed book (only allowed material: an A4 sheet with handwritten notes on both sides) in person exam. This tests a different form of understanding than open book take-home exams. While the latter focus on depth of understanding, the closed book in person exams focus on the immediate accessibility of the student's core knowledge (measured by the ability of answering basic questions under pressure). The exam will take place on Monday 12 October in the 2pm to 5pm assessment activity, as scheduled on MyTimetable.

In comparison to MATH3015 exam attempts, extra depth is expected from MATH6115 students. The arguments need to be more than globally correct; they need to address fine technical details.

Assessment Task 3

Oral presentation videos and discussion

Small group work including pre-recorded oral presentation, large group discussion, and quizzes on rapid information retention. The final seminar phase of the course uses a flipped classroom approach. This means that, instead of delivering live oral presentations, you will be asked to produce a 5-10 minute video, similar to the course video material (but much more condensed). These videos will be produced in groups. All students will need to watch all of the other students’ videos. The goal of the presentation is to maximise the learning of the audience (the rest of the class).

We will discuss the content during in person meetings, and the expectation is that everybody will contribute to discussions. These discussions take a transdisciplinary approach: they should include other perspectives than the perspective of professional mathematicians, including perspectives from other academic disciplines, as well as perspectives from industry. One goal is to understand what mathematicians have to say on the topics, highlighting the strengths but also the limitations of their perspective.

The discussion of the oral presentation videos will take place in Weeks 11-12. Please note that the transdisciplinary perspective of MATH6115 students is expected to be strong and demonstrate deep understanding of the perspectives of stakeholders who are not academic mathematicians.

Assessment Task 4

Essay

The final essay is a reflection letter whose recipient should be considered to be the lecturer or a mathematical colleague of the lecturer, demonstrating depth of thought process in approaching complex problems. This is a 5 page document related to the oral presentation project, written for a specific reader. It needs to tell a story, explaining how the student personally thinks of different theorems, adding details to a proof that the student personally needed to convince themselves of, or working out examples that helped to understand the material. The essay mark will measure the added value of the student's human contribution, compared to what an AI would produce.

In comparison to MATH3015 essay submissions, the transdisciplinary perspective of MATH6115 students is expected to be strong and demonstrate deep understanding of the perspectives of stakeholders who are not academic mathematicians.

Academic Integrity

Academic integrity is a core part of our culture as a community of scholars. At its heart, academic integrity is about behaving ethically. This means that all members of the community commit to honest and responsible scholarly practice and to upholding these values with respect and fairness. The Australian National University commits to embedding the values of academic integrity in our teaching and learning. We ensure that all members of our community understand how to engage in academic work in ways that are consistent with, and actively support academic integrity. The ANU expects staff and students to uphold high standards of academic integrity and act ethically and honestly, to ensure the quality and value of the qualification that you will graduate with. The University has policies and procedures in place to promote academic integrity and manage academic misconduct. Visit the following Academic honesty & plagiarism website for more information about academic integrity and what the ANU considers academic misconduct. The ANU offers a number of services to assist students with their assignments, examinations, and other learning activities. The Academic Skills and Learning Centre offers a number of workshops and seminars that you may find useful for your studies.

Online Submission

See Canvas. MATH6115 does not use Turnitin, having been granted an exemption.

Hardcopy Submission

For some forms of assessment (hand written assignments, art works, laboratory notes, etc.) hard copy submission is appropriate when approved by the Associate Dean (Education). Hard copy submissions must utilise the Assignment Cover Sheet. Please keep a copy of tasks completed for your records.

Late Submission

Late submission of assessment tasks without an extension are penalised at the rate of 5% of the possible marks available per working day or part thereof. Late submission of assessment tasks is not accepted after 10 working days after the due date, or on or after the date specified in the course outline for the return of the assessment item. 

Referencing Requirements

Accepted academic practice for referencing sources that you use in presentations can be found via the links on the Wattle site, under the file named “ANU and College Policies, Program Information, Student Support Services and Assessment”. Alternatively, you can seek help through the Students Learning Development website.

Returning Assignments

Marked pieces of assessment are returned to the student electronically.

Extensions and Penalties

Extensions and late submission of assessment pieces are covered by the Student Assessment (Coursework) Policy and Procedure The Course Convener may grant extensions for assessment pieces that are not examinations or take-home examinations. If you need an extension, you must request an extension in writing on or before the due date. If you have documented and appropriate medical evidence that demonstrates you were not able to request an extension on or before the due date, you may be able to request it after the due date.

Resubmission of Assignments

Resubmission of assignments is possible up to the submission deadline, but not after.

Privacy Notice

The ANU has made a number of third party, online, databases available for students to use. Use of each online database is conditional on student end users first agreeing to the database licensor’s terms of service and/or privacy policy. Students should read these carefully. In some cases student end users will be required to register an account with the database licensor and submit personal information, including their: first name; last name; ANU email address; and other information. In cases where student end users are asked to submit ‘content’ to a database, such as an assignment or short answers, the database licensor may only use the student’s ‘content’ in accordance with the terms of service — including any (copyright) licence the student grants to the database licensor. Any personal information or content a student submits may be stored by the licensor, potentially offshore, and will be used to process the database service in accordance with the licensors terms of service and/or privacy policy. If any student chooses not to agree to the database licensor’s terms of service or privacy policy, the student will not be able to access and use the database. In these circumstances students should contact their lecturer to enquire about alternative arrangements that are available.

Distribution of grades policy

Academic Quality Assurance Committee monitors the performance of students, including attrition, further study and employment rates and grade distribution, and College reports on quality assurance processes for assessment activities, including alignment with national and international disciplinary and interdisciplinary standards, as well as qualification type learning outcomes. Since first semester 1994, ANU uses a grading scale for all courses. This grading scale is used by all academic areas of the University.

Support for students

The University offers students support through several different services. You may contact the services listed below directly or seek advice from your Course Convener, Student Administrators, or your College and Course representatives (if applicable).

• ANU Health, safety & wellbeing for medical services, counselling, mental health and spiritual support
• ANU Diversity and inclusion for students with a disability or ongoing or chronic illness
• ANU Dean of Students for confidential, impartial advice and help to resolve problems between students and the academic or administrative areas of the University
• ANU Academic Skills and Learning Centre supports you make your own decisions about how you learn and manage your workload.
• ANU Counselling Centre promotes, supports and enhances mental health and wellbeing within the University student community.
• ANUSA supports and represents undergraduate and ANU College students
• PARSA supports and represents postgraduate and research students