University of Technology, Sydney

Staff directory | Webmail | Maps | Newsroom | What's on

25875 Probability Theory and Stochastic Analysis

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 2020 is available in the Archives.

UTS: Science: Mathematical and Physical Sciences
Credit points: 8 cp

Subject level:

Postgraduate

Result type: Grade and marks

There are course requisites for this subject. See access conditions.

Description

The subject introduces students to a number of fundamental results and techniques from probability theory and stochastic analysis that are used extensively for derivative pricing and hedging, interest rate modelling, and risk management.

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.

Contribution to the development of graduate attributes

1. Disciplinary knowledge and its appropriate application

An understanding of the nature, practice & application of the chosen science discipline (Assessment Tasks 1, 2 & 3).

2. An Inquiry-oriented approach

Encompasses problem solving, critical thinking and analysis attributes and an understanding of the scientific method of knowledge acquisition (Assessment Tasks 1, 2 & 3).

3. Professional skills and their appropriate application

The ability to acquire, develop, employ and integrate a range of technical, practical and professional skills, in appropriate and ethical ways within a professional context, autonomously and collaboratively and across a range of disciplinary and professional area; e.g. time management skills, personal organisation skills, teamwork skills, computing skills, laboratory skills, data handling, quantitative and graphical literacy skills (Assessment Tasks 1, 2 & 3).

6. Communication skills

An understanding of the different forms of communication - writing, reading, speaking, listening - including visual and graphical, within science and beyond and the ability to apply these appropriately and effectively for different audiences (Assessment Tasks 1, 2 & 3).

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 and its appropriate application

2. An Inquiry-oriented approach

3. Professional skills and their appropriate application

6. Communication skills.

Objective(s):

This assessment task addresses subject learning objective(s):

1, 2 and 3

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 and its appropriate application

2. An Inquiry-oriented approach

3. Professional skills and their appropriate application

6. Communication skills.

Objective(s):

This assessment task addresses subject learning objective(s):

1, 2 and 3

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 and its appropriate application

2. An Inquiry-oriented approach

3. Professional skills and their appropriate application

6. Communication skills.

Type: Examination
Groupwork: Individual
Weight: 40%
Criteria:

Application of appropriate theoretical content, accuracy of analysis, clarity of communication of solutions.

Minimum requirements

Students must achieve at least 50% of the subject’s total marks.

References

Platen, E. & Heath, D.: A Benchmark Approach to Quantitative Finance
Springer Finance, 700 pp., 199 illus., Hardcover, ISBN-10 3-540-26212-1 (2006)

Additional references

Karatzas, I., and Shreve, S. E., Brownian Motion and Stochastic Calculus, 2nd edition, Springer, 1991
Oksendal, B., Stochastic Differential Equations: An Introduction with Applications, 6th edition, Springer, 2010