# SF2522 Computational Methods for Stochastic Differential Equations 7.5 credits

Advanced course on modeling, analysis and effecient computation of solutions of problems with stochastic phenomena in science and technology.

The course contains the basic mathematical theory of stochastic differential equations and optimal control as well as its application to real problems in financial mathematics, material science, fluid flow, broadcast networks, optimal design, optimal reconstruction and chemical reactions in cell biology.

Course offering missing for current semester as well as for previous and coming semesters
Headings with content from the Course syllabus SF2522 (Spring 2020–) are denoted with an asterisk ( )

## Content and learning outcomes

### Course contents

The course treats stochastic differential equations and their numerical solution, with applications in financial mathematics, turbulent diffusion, control theory and Monte Carlo methods. We discuss basic questions for solving stochastic differential equations, e.g. to determine the price of an option is it more efficient to solve the deterministic Black and Scholes partial differential equation or use a Monte Carlo method based on stochastics.

The course treats basic theory of stochastic differential equations including weak and strong approximation, efficient numerical methods and error estimates, the relation between stochastic differential equations and partial differential equations, stochastic partial differential equations, variance reduction.

### Intended learning outcomes

After completing this master level course the students will be able to model, analyze and efficiently compute solutions to problems including random phenomena in science and engineering. The student learns the basic mathematical theory for stochastic differential equations and optimal control and applies it to some real-world applications, including financial mathematics, material science, geophysical flow, radio networks, optimal design, optimal reconstruction, and chemical reactions in cell biology.

More precisely the goal of the course means that the student can:

• present some models in science and finance based on stochastic differential equations and evaluate methods to determine their solution,
• derive and use the correspondence between expected values of stochastic diffusion processes and solutions to certain deterministic partial differential equations,
• formulate, use and analyze the main numerical methods for stochastic differential equations, based on Monte Carlo stochastics and partial differential equations,
• present some stochastic and deterministic optimal control problems in science and finance using differential equations and Markov chains,
• formulate, use and analyze deterministic and stochastic optimal control problems using both differential equations constrained minimization and dynamic programming (leading to the Hamilton-Jacobi-Bellman nonlinear partial differential equation),
• derive the Black-Scholes equation for options in mathematical finance and analyze the alternatives to determine option prices numerically,
• determine and analyze reaction rate problems for stochastic differential equation with small noise using optimal control theory.

### Course disposition

No information inserted

## Literature and preparations

### Specific prerequisites

The prerequisite for the course is linear algebra, calculus, differential equations, probability and numerics corresponding to the first three years at KTH.

### Recommended prerequisites

No information inserted

### Equipment

No information inserted

### Literature

No information inserted

## Examination and completion

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

A, B, C, D, E, FX, F

### Examination

• LABA - Laboratory, 3.5 credits, grading scale: P, F
• TEN1 - Examination, 4.0 credits, grading scale: A, B, C, D, E, FX, 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.

In this course all the regulations of the code of honor at the School of Computer science and Communication apply, see: http://www.kth.se/csc/student/hederskodex/1.17237?l=en_UK.

### Other requirements for final grade

The course includes lectures, collaborative homework problems and student presentations of their computations inspired by recent research papers.
The homework and projects give credit points for a final written exam.

### 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 web

Further information about the course can be found on the Course web at the link below. Information on the Course web will later be moved to this site.

Course web SF2522

SCI/Mathematics

### Main field of study

Mathematics, Technology

### Education cycle

Second cycle

No information inserted

### Contact

Anders Szepessy (szepessy@kth.se)

### Transitional regulations

Written examination:

Four re-exams are offered for two years after the last course edition was given. The examinations in the course SF2522 are given together with those of the course SF2525 Computational Methods for Stochastic Differential Equations and Machine Learning 7,5 credits that replaces SF2522. The first two re-examinations have been given during spring semester 2020.

Approved oral and/or written presentation of homeworks

Lab A can be finalized within the new course SF2525. The deadlines for lab work/project work of the course SF2525 are also valid for students who intend to finalize Lab A in the course SF2522.

### Supplementary information

The course is given every second year.