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Syllabus:

# Stochastic Differential Equations, 7.5 Credits

Swedish name: Stokastiska differentialekvationer

This syllabus is valid: 2024-07-22 and until further notice

Course code: 5MA180

Credit points: 7.5

Education level: Second cycle

Main Field of Study and progress level: Mathematics: Second cycle, has only first-cycle course/s as entry requirements
Computational Science and Engineering: Second cycle, has only first-cycle course/s as entry requirements
Mathematical Statistics: Second cycle, has only first-cycle course/s as entry requirements

Established by: Faculty Board of Science and Technology, 2017-10-01

Revised by: Faculty Board of Science and Technology, 2024-02-20

## Contents

Module1 (6.5 hp): Theory.
The module starts with a review of the necessary prerequisites in probability theory, including an introduction to measure theory and stochastic processes . Thereafter (local) martingales and the quadratic variation are introduced with its most famous example being the Brownian motion.  The Ito integral and the Ito calculus are introduced.,  This is applied to solving certain stochastic differential equations (SDE) analytically . Furthermore, the existence- and uniqueness theory for SDE is treated in the Lipschitz case, which naturally leads to numerical methods for simulating solutions to SDEs. The connection between SDE and partial differential equations (PDE) is investigated (e.g.,  the Feynman-Kac equation), which gives the possibility to simulate solutions of PDEs in separate points by using simulations of SDEs.  Additionally, Girsanov's theorem and the martingale representation theorem are discussed, as well as a quick introduction to optimal stopping problems.

Module 2 (1 hp) Computer labs.
The module covers implementation of some numerical method for simulating solutions, fitting model parameters to given data, and the Least-Square-Monte-Carlo (LSMC) method for solving optimal stopping problems.

## 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

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

•  solve challenging numerical problem with the techniques acquired in the course, and evaluate the models with a scientific perspectives,

## 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  a numerical group project.

## Examination modes

Module 1 is assessed through written and oral 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.

Examiners may decide to deviate from the modes of assessment in the course syllabus. Individual adaption of modes of assessment must give due consideration to the student's needs. The adaption of modes of assessment must remain within the framework of the intended learning outcomes in the course syllabus. Students who require an adapted examination must submit a request to the department holding the course no later than 10 days before the examination. The examiner decides on the adaption of the examination, after which the student will be notified.

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.

In the event that the syllabus ceases to apply or undergoes major changes, students are guaranteed at least three examinations (including the regular examination opportunity) according to the regulations in the syllabus that the student was originally registered on for a period of a maximum of two years from the time that the previous syllabus ceased to apply or that the course ended.

## Literature

### Valid from: 2024 week 30

Øksendal Bernt
Stochastic differential equations : an introduction with applications
3. ed. : Berlin : Springer : cop. 1992 : xiii, 224 s. :
ISBN: 3540533354
Mandatory
Search the University Library catalogue

Pascucci Andrea.
PDE and Martingale Methods in Option Pricing
Milano : Springer Milan : 2011 :
ISBN: 9788847017818
Mandatory
Search the University Library catalogue