The course offers an introduction to modern stochastic processes, including Brownian motion, continuous-time martingales, stochastic integration and Ito's calculus, Markov processes, stochastic differential equations, point processes and their applications. The course will include some applications but will emphasise setting up a solid theoretical foundation for the subject.

The course will provide a sound basis for progression to other honours and post-graduate courses including mathematical finance, stochastic analysis and statistics, and actuarial sciences.

The course aims to round off the rigorous introduction to probabilistic reasoning initiated in STAT7018, as well as to substantially enhance students' depth of knowledge in the mathematical underpinning of stochastic process theory. 

Learning Outcomes

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

On satisfying the requirements of this course, students will have the knowledge and skills to:
1. Explain in detail the fundamental concepts of stochastic processes in  continuous time and their position in modern statistical and mathematical sciences and applied contexts;
2. Demonstrate to a high level accurate and efficient use of stochastic calculus techniques;
3. Demonstrate and be able to communicate capacity for mathematical reasoning through analyzing, proving and explaining concepts from stochastic analysis;
4. Apply problem-solving using stochastic analysis methods in various situations.

Research-Led Teaching

ANU has a rich history of research in the area of applied probability and stochastic processes . The lecturer and other RSFAS staff members are active researchers in this area, with a keen interest to attract talented students for research projects.

Examination Material or equipment

Two sheets of A4 paper with personal annotations on both sides; paper-based dictionary, no approval (must be clear of all annotations); calculator (non-programmable).

Required Resources

Course and Lecture Notes will be made available on WATTLE.

Recommended Resources

The following books are made available in the two days reserve of the Hancock library. 

Preliminary Reading (STAT7018): Rosenthal, Jeffrey S. (2008) A first look at rigorous probability theory. 2nd edition, World Scientific, Singapore.

Recommended Reading: Klebaner, Fima C. (2012) Introduction to stochastic calculus with applications. Third Edition. Imperial College Press, London.

Revusz, Daniel; Yor, Marc (1999) Continuous martingales and Brownian motion. 3rd edition. Grundlehren der Mathematischen Wissenschaften 293, Springer-Verlag, Berlin.

Steele, J. Michael (2001) Stochastic calculus and financial applications. New York : Springer, c2001

Staff Feedback

Students will be given feedback in the following forms in this course:

• Written comments
• Verbal comments
• Feedback to the whole class

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.

Class Schedule

Week/Session	Summary of Activities	Assessment
1	Conditional expectation, Markov chains and martingales
2	Conditional expectation, Markov chains and martingales
3	Conditional expectation, Markov chains and martingales
4	Brownian motion: existence, path properties, as Markov process, as martingale, reflection principle, hitting and exit times, multivariate Brownian notion.
5	Brownian motion: existence, path properties, as Markov process, as martingale, reflection principle, hitting and exit times, multivariate Brownian notion.	Assignment 1 due
6	Brownian motion calculus: Ito integration, Ito formula, Ito’s formula for Ito processes, Stratonovich integral.
7	Brownian motion calculus: Ito integration, Ito formula, Ito’s formula for Ito processes, Stratonovich integral.
8	Stochastic differential equations: definition, stochastic exponential and logarithm, existence and uniqueness of strong solutions, Markov property, weak solutions, backward and forward equations.	Assignment 2 due
9	Stochastic differential equations: definition, stochastic exponential and logarithm, existence and uniqueness of strong solutions, Markov property, weak solutions, backward and forward equations.
10	Diffusions: martingales and Dynkin formula, expectations, PDEs, exit times, representations of ODEs, explosion, recurrence transience, diffusions on an interval, stationary distributions, multivariate diffusions.
11	Diffusions: martingales and Dynkin formula, expectations, PDEs, exit times, representations of ODEs, explosion, recurrence transience, diffusions on an interval, stationary distributions, multivariate diffusions.	Assignment 3 due
12	Optional topics: Point processes, Levy processes, Applications to finance, biology or engineering.

Tutorial Registration

Tutorial signup for this course will be done via the Wattle website.

Detailed information about signup times will be provided on Wattle.

When tutorials are available for enrolment, follow these steps:

1. Log on to Wattle, and go to the course site

2. Click on the link “Tutorial enrolment”

3. On the right of the screen, click on the tab “Become Member of…..” for the tutorial class you wish to enter

4. Confirm your choice If you need to change your enrolment, you will be able to do so by clicking on the tab “Leave group….” and then re-enrol in another group. You will not be able to enrol in groups that have reached their maximum number. Please note that enrolment in ISIS must be finalised for you to have access to Wattle.

Assessment Summary

Assessment task	Value	Due Date	Return of assessment	Learning Outcomes
Assignment 1	12 %	16/08/2019	23/08/2019	1-4
Assignment 2	16 %	27/09/2019	04/10/2019	1-4
Assignment 3	12 %	18/10/2019	25/10/2019	1-4
Final Examination	60 %	31/10/2019	28/11/2019	1-4

* 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.

Examination(s)

Final Examination is compulsory. Further information about the Final Examination will be provided in class and on Wattle closer to the exam date. Centrally administered examinations through Examinations, Graduations & Prizes will be timetabled prior to the examination period. Please check ANU Timetabling for further information. Exam scripts will not be returned.

Assessment Task 1

Assignment 1

Problem sheets are published on a weekly basis via WATTLE on the same day of the second lecture each week. Each problem sheet contains tutorial and homework questions. Assignment 1 comprises submission of all homework questions released for weeks 1-3 on the due date. You may type your answer in a type-setting software or you may hand-write parts of your answers. Please ensure that your handwriting is legible. Assignments are due at 23:59 on the due date.

Assessment Task 2

Assignment 2

Problem sheets are published on a weekly basis via WATTLE on the same day of the second lecture each week. Each problem sheet contains tutorial and homework questions. Assignment 1 comprises submission of all homework questions released for weeks 4-7 on the due date. You may type your answer in a type-setting software or you may hand-write parts of your answers. Please ensure that your handwriting is legible. Assignments are due at 23:59 on the due date.

Assessment Task 3

Assignment 3

Problem sheets are published on a weekly basis via WATTLE on the same day of the second lecture each week.Each problem sheet contains tutorial and homework questions. Assignment 1 comprises submission of all homework questions released for weeks 8-10 on the due date. You may type your answer in a type-setting software or you may hand-write parts of your answers. Please ensure that your handwriting is legible. Assignments are due at 23:59 on the due date.

Assessment Task 4

Final Examination

Final Examination is compulsory. Further information about the Final Examination will be provided in class and on Wattle closer to the exam date. Centrally administered examinations through Examinations, Graduations & Prizes will be timetabled prior to the examination period. Please check ANU Timetabling for further information. Exam scripts will not be returned.

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

The ANU uses 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. While the use of Turnitin is not mandatory, the ANU highly recommends Turnitin is used by both teaching staff and students. For additional information regarding Turnitin please visit the ANU Online website.

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

No submission of assessment tasks without an extension after the due date will be permitted. If an assessment task is not submitted by the due date, a mark of 0 will be awarded.

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

Assignments will be returned in class.

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 not allowed.

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