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Mathematical Sciences Institute · ANU College of Systems and Society
This course gives a rigorous mathematical introduction to stochastic processes and their calculus, stochastic differential equations, and their applications in finance. The first half of the course covers martingales, Brownian motion, Ito integration, stochastic differential equations driven by a Brownian motion, and their applications to pricing European options using the Black-Scholes model. The second half of the course covers a range of more advanced topics to be selected by the students. This includes change of measure theory, its application to pricing complex options, relationship between stochastic analysis and partial differential equations (and its computational uses), basics of Malliavin stochastic calculus of variations and its application in risk management, as well as topics in statistics (such as model fitting), economics (such as interest rates modelling) or other disciplines (such as generative AI in computer science, quantum field theory in physics, or insect behaviour in biology). The first part of the course aims to develop a common perspective and language, anchored in mathematical analysis, and building on a large part of the undergraduate curriculum in Mathematics (calculus, differential equations, analysis in metric spaces, measure theory, probability theory). In the second part of the course, students develop an ability to use their mathematical knowledge in collaborative transdisciplinary complex problem solving. This is done by working in groups on projects that are too complex to be fully completed in a short period of time, and with collaborators who have some common language, but different backgrounds (typically some collaborators with a primary training in mathematics, and some with a primary training in finance or economics). The projects highlight the different modes of communication and value systems of its potential stakeholders, including academics from different disciplines, as well as practitioners from different industries (e.g. what makes an argument convincing to a mathematician vs an economist, and what kind of solutions need to be achieved in investment banking vs in government). The course operates as a flipped classroom, with technical content delivered through videos and lecture notes, and in person activities focusing on discussions. These discussions aim to develop communication skills as well as critical thinking skills. They involve technical discussions of the key parts of proofs and the nature of assumptions in theorems. They also involve historical and societal perspectives on the development of the ideas expressed through the theorems and their proofs, as well as their use in complex problems. The course provides a sound foundation for progress to honours and post-graduate courses in probability, analysis, or differential equations. It also provides both technical and communication training for working in a range of professions (such as quantitative analyst or data scientist in government, or in finance, insurance, or information technology industries). Note: This is an Honours Pathway Course. It continues the development of sophisticated mathematical techniques and their application begun in MATH3029 or MATH3320.
Regular written assignments
Learning outcomes 1, 2, 3
Mid-semester examination
Learning outcomes 1, 2, 3
Small group work including pre-recorded oral presentation, large group discussion, and quizzes on rapid information retention
Learning outcomes 1, 2, 3, 4, 5
Final essay as a reflection letter to the lecturer, demonstrating depth of thought process in approaching complex problems
Learning outcomes 1, 2, 3, 5
The expected workload will consist of approximately 130 hours throughout the semester. The course operates as a flipped classroom. This means that students first learn the content before meeting with the lecturer, and then they use the in-person class time to reach a deep level of understanding. In practice, this means the expected workload is: • Approximately 82 hours of self-directed study which will include: - watching approximately 33 pre-recorded videos and reading lecture notes covering the course content, to prepare for the in-person activities in Weeks 1 to 10, - preparation for assessment tasks, and - watching through other students' oral presentation videos in preparation for the discussions of them in Weeks 11 and 12; • Face-to face component which will consist of approximately 48 hours of scheduled in-person activities during the semester. These in-person activities will include a 1 hour workshop most weeks plus 3 hours per week of other in person activities (detail in the class summary).
There is no prescribed textbook. Extensive lecture notes are provided. Preliminary Reading "Stochastic calculus and financial applications" by M. Steele (ISBN 978-1-4684-9305-4) is highly recommended.