• Svenska

# FSF3713 Stochastic Analysis 7.5 credits

### For course offering

Spring 2024 Start 16 Jan 2024 programme students

### Target group

PhD students only.

### Part of programme

No information inserted

### Periods

P3 (3.0 hp), P4 (4.5 hp)

16 Jan 2024
3 Jun 2024

25%

Normal Daytime

English

KTH Campus

### Number of places

Places are not limited

## Application

### For course offering

Spring 2024 Start 16 Jan 2024 programme students

60782

## Contact

### For course offering

Spring 2024 Start 16 Jan 2024 programme students

### Contact

Fredrik Viklund (frejo@kth.se)

### Examiner

No information inserted

### Course coordinator

No information inserted

### Teachers

No information inserted
Headings with content from the Course syllabus FSF3713 (Spring 2019–) are denoted with an asterisk ( )

## Content and learning outcomes

### Course contents

• Stochastic processes, martingales, local martingales, stopping times, filtrations, Markov properties.
• Brownian motion.
• The Ito isometry, Ito integrals, Ito’s formula.
• Existence and uniqueness of solutions to stochastic differential equations.
• Diffusion processes.
• Girsanov’s theorem.
• Probabilistic representations of solutions to partial differential equations.
• The Feynman­Kac formula, Kolmogorov’s forward and backward equations, recurrence, invariant densities.
• More advanced topics, e.g., local times, if time permits.

### Intended learning outcomes

• Understand and explain the following concepts: filtration, stochastic process in continuous time, local martingale, martingale, stopping time, quadratic variation.
• Sketch at least one construction of Brownian motion.
• Construct the Ito integral in some generality, and discuss its basic properties.
• Explain Ito’s formula and use it for practical computations of, e.g., Ito integrals.
• Discuss basic properties of stochastic differential equations (SDEs), in particular diffusions in one dimension.
• Explain Girsanov’s theorem.
• Discuss connections between the theory of SDEs and partial differential equations.
• Solve problems and discuss current research connected to the theory presented in the course

## Literature and preparations

### Specific prerequisites

Basic probability theory (e.g., SF3940) and basic knowledge in analysis and linear algebra, especially measure theory and Lebesgue integration.

### Recommended prerequisites

No information inserted

### Equipment

No information inserted

### Literature

For example

Karatzas­Shreve “Brownian motion and stochastic calculus” (ISBN 978-1-4612-­0949-­2),

Revuz­Yor “Continuous martingales and Brownian motion” (ISBN 978­-3-662­-06400-­9);

Öksendal “Stochastic Differential Equations” (ISBN 978-­3-­642­14394­-6)

## Examination and completion

If the course is discontinued, students may request to be examined during the following two academic years.

P, F

### Examination

• INL1 - Assignment, 3.5 credits, grading scale: P, F
• TENM - Oral exam, 4.0 credits, grading scale: P, F

Based on recommendation from KTH’s coordinator for disabilities, the examiner will decide how to adapt an examination for students with documented disability.

The examiner may apply another examination format when re-examining individual students.

Homework assignments and oral/written exam.

### Other requirements for final grade

Homework assignments completed and oral/written exam passed.

### Opportunity to complete the requirements via supplementary examination

No information inserted

### Opportunity to raise an approved grade via renewed examination

No information inserted

### Ethical approach

• All members of a group are responsible for the group's work.
• In any assessment, every student shall honestly disclose any help received and sources used.
• In an oral assessment, every student shall be able to present and answer questions about the entire assignment and solution.

## Further information

### Course room in Canvas

Registered students find further information about the implementation of the course in the course room in Canvas. A link to the course room can be found under the tab Studies in the Personal menu at the start of the course.

### Main field of study

This course does not belong to any Main field of study.

### Education cycle

Third cycle

No information inserted

### Contact

Fredrik Viklund (frejo@kth.se)

### Supplementary information

PhD students only.