Undergraduate Course: Stochastic Differential Equations (MATH10085)
Course Outline
| School | School of Mathematics |
College | College of Science and Engineering |
| Credit level (Normal year taken) | SCQF Level 10 (Year 4 Undergraduate) |
Availability | Available to all students |
| SCQF Credits | 10 |
ECTS Credits | 5 |
| Summary | Stochastic methods, stochastic differential equations (SDEs) in particular, are used extensively in finance, industry and in sciences. Reflecting this, this course provides an introduction to SDEs that discusses the fundamental concepts and properties of SDEs and presents strategies for their exact, approximate, and numerical solution. The first part of the course focuses on theoretical concepts, including the definition of Brownian motion and stochastic integrals, and on analytical techniques for the solution of SDEs; it also emphasises the connections between SDEs and partial differential equations. The second part centres on numerical methods for both strong and weak approximations of solutions and introduces widely used numerical schemes. |
| Course description |
Part 1: Introduction to SDEs
- Brownian motion: random walks, Wiener process, white noise.
- Stochastic integrals: definition, Ito isometry, Ito¿s formula
- SDEs: definitions, existence and uniqueness, examples
- Applications: applications to PDEs (Laplace equation, Feynman-Kac), limit of coloured noise (Stratonovich SDEs and conversion rules).
Part 2: Numerical SDEs
- Strong and weak approximations of solutions to SDEs,
- Euler approximations, Milstein scheme,
- Order of accuracy of the approximations,
- Higher order schemes, accelerated convergence,
- Weak approximations of SDEs via numerical solutions of PDEs.
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Information for Visiting Students
| Pre-requisites | None |
Course Delivery Information
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| Academic year 2014/15, Available to all students (SV1)
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Quota: None |
| Course Start |
Semester 2 |
Timetable |
Timetable |
| Learning and Teaching activities (Further Info) |
Total Hours:
100
(
Lecture Hours 22,
Seminar/Tutorial Hours 5,
Summative Assessment Hours 2,
Programme Level Learning and Teaching Hours 2,
Directed Learning and Independent Learning Hours
69 )
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| Assessment (Further Info) |
Written Exam
95 %,
Coursework
5 %,
Practical Exam
0 %
|
| Additional Information (Assessment) |
Coursework 5%, Examination 95% |
| Feedback |
Not entered |
| No Exam Information |
Learning Outcomes
Understanding the concepts of Brownian motion and white noise.
Ability to manipulate and solve simple SDEs.
Understanding the relationship between SDEs and parabolic PDEs.
Understanding the concept of strong and weak approximations to solutions of SDEs.
Familiarity with standard numerical algorithms for the solution of SDEs.
Appreciation of the challenges posed by accurate numerical solutions of SDEs.
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Reading List
| L C Evans, An introduction to stochastic differential equations, AMS (2013). |
Additional Information
| Graduate Attributes and Skills |
Not entered |
| Keywords | SDE |
Contacts
| Course organiser | Dr Jacques Vanneste
Tel: (0131 6)50 6483
Email: J.Vanneste@ed.ac.uk |
Course secretary | Mrs Alison Fairgrieve
Tel: (0131 6)50 5045
Email: Alison.Fairgrieve@ed.ac.uk |
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