Undergraduate Course: Stochastic Differential Equations (MATH10085)
Course Outline
| School | School of Mathematics |
College | College of Science and Engineering |
| Course type | Standard |
Availability | Available to all students |
| Credit level (Normal year taken) | SCQF Level 10 (Year 4 Undergraduate) |
Credits | 10 |
| Home subject area | Mathematics |
Other subject area | None |
| Course website |
None |
Taught in Gaelic? | No |
| Course description | 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. |
Information for Visiting Students
| Pre-requisites | None |
| Displayed in Visiting Students Prospectus? | Yes |
Course Delivery Information
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| Delivery period: 2014/15 Semester 2, Available to all students (SV1)
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Learn enabled: Yes |
Quota: None |
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Web Timetable |
Web Timetable |
| Course Start Date |
12/01/2015 |
| Breakdown of 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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| Additional Notes |
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| Breakdown of Assessment Methods (Further Info) |
Written Exam
95 %,
Coursework
5 %,
Practical Exam
0 %
|
| No Exam Information |
Summary of Intended 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. |
Assessment Information
| Coursework 5%, Examination 95% |
Special Arrangements
| None |
Additional Information
| Academic description |
Not entered |
| Syllabus |
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. |
| Transferable skills |
Not entered |
| Reading list |
L C Evans, An introduction to stochastic differential equations, AMS (2013). |
| Study Abroad |
Not entered |
| Study Pattern |
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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