37464 Advanced Stochastic Processes
Warning: The information on this page is indicative. The subject outline for a
particular session, location and mode of offering is the authoritative source
of all information about the subject for that offering. Required texts, recommended texts and references in particular are likely to change. Students will be provided with a subject outline once they enrol in the subject.
Subject handbook information prior to 2024 is available in the Archives.
Credit points: 6 cp
Result type: Grade and marks
Requisite(s): 35361 Stochastic Processes OR 37363 Stochastic Processes and Financial Mathematics
These requisites may not apply to students in certain courses.
There are course requisites for this subject. See access conditions.
Anti-requisite(s): 35466 Advanced Stochastic Processes
Description
This subject aims to introduce honours students to the mathematical theory and some financial applications of Brownian motion and related processes. It covers the following topics: formal definition of probability space and stochastic processes; Martingales; Riemann-Stieltjes integration; Brownian motion and related processes; stochastic calculus and stochastic differential equations; and financial applications.
Subject learning objectives (SLOs)
Upon successful completion of this subject students should be able to:
| 1. | Understand the main concepts and classifications of stochastic processes |
|---|---|
| 2. | Derive the main results in stochastic analysis. |
| 3. | Formulate and solve theoretical and applied problems (with an emphasis on finance and mathematical statistics) using the analytical and numerical approaches of stochastic analysis. |
| 4. | Relate the assumptions and limitations of the tools used to solve applied problems in stochastic analysis. |
| 5. | Communicate the solutions and results of applied problems in written language accessible to non-specialists. |
Course intended learning outcomes (CILOs)
This subject also contributes specifically to the development of following course intended learning outcomes:
- Apply: Develop a well-developed broad range of mathematical, statistical, computational, and data management skills, as well as experience in the use of the information technology required for modern data analysis. (1.1)
- Analyse: Make arguments based on proof and conduct simulations based on selection of approaches (e.g. analytic vs numerical/experimental, different statistical tests, different heuristic algorithms) and various sources of data and knowledge. (2.2)
- Analyse: Organise and manage a complex project demonstrating advanced skills in Mathematical Programming and Specialist Mathematical/Statistical/QM Software using time management and collaborative skills. (3.2)
- Apply: Succinct and accurate presentation of information, reasoning, and conclusions in a variety of modes to diverse expert and non-expert audiences. (5.1)
Contribution to the development of graduate attributes
The material presented in this subject and the method of presentation are linked to the following Science graduate attributes:
1. Disciplinary knowledge.
Knowledge of mathematical sciences to demonstrate depth, breadth, application, and interrelationships of relevant discipline areas.
2. Research, inquiry and critical thinking.
The ability to frame conjectures and hypotheses using a scientific approach, to test current mathematics knowledge through critical evaluation and data analyses, and to solve problems through theoretical work and/or experimental observation.
3. Professional, ethical, and social responsibility.
A capacity to work ethically and professionally using technical, practical, and collaborative mathematical skills within the context of the workplace, and apply these to meet the current and future needs of society.
5. Communication.
Effective and professional communication skills for a range of scientific audiences.
Teaching and learning strategies
Classes each week will comprise a two-hour seminar combining lecture and tutorial components. This will be complemented by at least three hours of individual work involving the review of the theoretical material, solution of theoretical and applied problems and implementation in computational software.
The subject outline, theoretical material, tutorial problems and assignment questions will be distributed in class and made available through UTSOnline.
Students will be required to attend and be prepared for each weekly session. Preparation must include learning the theoretical material of the preceding week and attempting the solution of all mathematical problems of the preceding week. This is critical because mathematics is learnt progressively, with mastery of the current step necessary before proceeding to the next. In this subject, learning the theoretical material from any given week will require the lessons learned from the previous week; mathematical problems from any given week will use tools developed from those used in the previous week.
Students will also be expected to actively participate in the presentation of small sections of the theoretical material. These sections will be allocated to groups (or individuals, depending on class size) and the group will collaborate in developing a small presentation and teaching this section of material to the rest of the class. This will be very such an informal and interactive process, with feedback and comments sought from the rest of the class, with the lecturer correcting, emphasising and filling in the gaps where necessary.
The same approach will be taken with respect to example problems. These will, similarly, be allocated to groups (or individuals, depending on class size), with each group responsible working through the problems, formulating solutions and talking the rest of the class through their methodology. There is no better way to learn a topic or a technique than to teach it to others.
In Week 1 the lecturer will provide details of consultation times, during which students may request assistance that could not be provided during formal class sessions. Questions may also be asked via email (addresses listed at top of document), with responses to be provided within two working days.
Content (topics)
The applications of this subject are wide and diverse, with recent focus on quantitative finance and risk management. However, the growing demand for numerical tools in biology is seeing new applications in genetic engineering, equilibrium analysis, bio-system modelling etc.
The basic object of study are stochastic processes, and tools such as Ito’s Lemma, partial differential equations and Monte Carlo simulation are used to solve expectations of functions of these processes. These expectations (or integrals) may represent the price of financial derivatives for instance.
Assessment
Assessment task 1: Assignment 1
| Intent: | This assessment task contributes to the development of the following graduate attributes: 1. Disciplinary knowledge 2. Research, inquiry and critical thinking 3. Professional, ethical and social responsibility 5. Communication skills |
|---|---|
| Objective(s): | This assessment task addresses subject learning objective(s): 1, 3, 4 and 5 This assessment task contributes to the development of course intended learning outcome(s): 1.1, 2.2, 3.2 and 5.1 |
| Type: | Exercises |
| Groupwork: | Individual |
| Weight: | 30% |
| Criteria: | Application of appropriate theoretical content, accuracy of analysis, clarity of communication of solutions. |
Assessment task 2: Assignment 2
| Intent: | This assessment task contributes to the development of the following graduate attributes: 1. Disciplinary knowledge 2. Research, inquiry and critical thinking 3. Professional, ethical and social responsibility 5. Communication skills |
|---|---|
| Objective(s): | This assessment task addresses subject learning objective(s): 1, 3, 4 and 5 This assessment task contributes to the development of course intended learning outcome(s): 1.1, 2.2, 3.2 and 5.1 |
| Type: | Exercises |
| Groupwork: | Individual |
| Weight: | 30% |
| Criteria: | Application of appropriate theoretical content, accuracy of analysis, clarity of communication of solutions. |
Assessment task 3: Final Examination
| Intent: | This assessment task contributes to the development of the following graduate attributes: 1. Disciplinary knowledge 2. Research, inquiry and critical thinking 3. Professional, ethical and social responsibility 5. Communication skills |
|---|---|
| Objective(s): | This assessment task addresses subject learning objective(s): 1, 2, 3, 4 and 5 This assessment task contributes to the development of course intended learning outcome(s): 1.1, 2.2, 3.2 and 5.1 |
| Type: | Examination |
| Groupwork: | Individual |
| Weight: | 40% |
| Criteria: | Application of appropriate theoretical content, accuracy of analysis, clarity of communication of solutions. |
Minimum requirements
In order to pass this subject, a student must achieve a final result of 50% or more and achieve 40% or more on the
final examination. The final result is simply the sum of all the marks gained in each piece of assessment. Students who
obtain 50 marks or more but fail to score 40% or more on the final examination will be given an X grade (Fail).
Required texts
Lecture Notes
Recommended texts
Gut, Allan: Probability. A Graduate Course. Springer, 2005. You can download this book from UTS library website.
Borovkov, Konstantin: Elements of Stochastic Modelling.
World Scientific Publishing Company; the second edition, 2014.
Platen, Eckhard and Heath, David: A benchmark approach to quantitative finance . Berlin , Springer, 2006.
You can download this book from UTS library website
Advanced textbooks
Cont, Rama and Tankov, Peter: Financial Modelling with Jump Processes. Chapman & Hall, 2004.
You can download chapters of this book.
Shreve, Steve. Stochastic Calculus for Finance. II. Continuous-Time Models. Springer, 2004.