Stochastic Differential Equations, 7.5 Credits

Contents

Module1 (6.5 hp): Theory.
The module starts with a review of the necessary prerequisites in probability theory, including an introduction to stochastic processes in discrete and continuous time. Thereafter the Brownian motion (the Wiener process), the Ito integral and the Ito calculus, with which Ito integrals can be transformed and in some cases calculated analytically, are introduced, in the next step stochastic differential equations (SDE) are introduced and certain types of SDE are solve analytically with Ito calculus. Furthermore the general existence- and uniqueness theory for SDE is treated, which naturally leads to numerical methods for simulating solutions to SDEs. The connection between SDE and partial differential equations (PDE) is investigated (e.g., Fokker-Plancks's equation), which gives the possibility to simulate solutions of PDEs in separate points by using simulations of SDEs. Finally, how SDE models are formulated and fitted to given data, is studied in some examples of applications.

Module 2 (1 hp) Computer labs.
The module covers implementation of some numerical method for simulating solutions, and fitting model parameters to given data.

Expected learning outcomes

For a passing grade, the student must be able to

Knowledge and understanding

• thoroughly account for central notions in Ito calculus
• thoroughly account for numerical simulation methods for Ito integrals, and solutions of stochastic differential equations
• thoroughly account for connections between stochastic differential equations and partial differential equations

Skills

• independently solve some Ito integrals and stochastic differential equations analytically
• use numerical simulation methods for Ito integrals and solutions to stochastic differential equations
• formulate mathematical models using stochastic differential equations

Judgement and approach

• fit stochastic differential equations models to given data and evaluate the models with a scientific perspective

Required Knowledge

The course requires 90 ECTS including 22,5 ECTS in Calculus of which 7,5 ECTC in Multivariable Calculus and Differential Equations, a basic course in Linear Algebra minimum 7,5 ECTS and a basic course in Mathematical Statistics minimum 6 ECTS. Proficiency in English and Swedish equivalent to the level required for basic eligibility for higher studies.

Form of instruction

The teaching takes the form of lectures, lessons and supervised lab work.

Examination modes

Module 1 is assessed through written examination. Module 2 is assessed through written lab reports. On Module 1, one of the following judgements is awarded: Fail (U), Pass (G) or Pass with distinction (VG). On Module 2, one of the following grades is awarded: Fail (U) or Pass (G). For the whole course, one of the following grades is awarded: Fail (U), Pass (G) or Pass with distinction (VG). To pass the whole course, all modules must have been passed. The grade for the whole course is determined by the judgement given for Module 1, and it is only set once all compulsory modules have been assessed. Students have possibility to raise the grade from G to VG through optional hand-in assignments. The bonus points awarded for the assignments are only valid on the first two exams when the course is given.

A student who has been awarded a passing grade for the course cannot be reassessed for a higher grade. Students who do not pass a test or examination on the original date are given another date to retake the examination. A student who has sat two examinations for a course or a part of a course, without passing either examination, has the right to have another examiner appointed, provided there are no specific reasons for not doing so (Chapter 6, Section 22, HEO). The request for a new examiner is made to the Head of the Department of Mathematics and Mathematical Statistics. Examinations based on this course syllabus are guaranteed to be offered for two years after the date of the student's first registration for the course.

Credit transfer
All students have the right to have their previous education or equivalent, and their working life experience evaluated for possible consideration in the corresponding education at Umeå university. Application forms should be addressed to Student services/Degree evaluation office. More information regarding credit transfer can be found on the student web pages of Umeå university, http://www.student.umu.se, and in the Higher Education Ordinance (chapter 6). If denied, the application can be appealed (as per the Higher Education Ordinance, chapter 12) to Överklagandenämnden för högskolan. This includes partially denied applications

Other regulations

This course can not be included in a degree together with another course with similar contents. When in doubt, the student should consult the director of study at the department of mathematics and mathematical statistics.