KTH Mathematics  


Mathematical Statistics

Course objectives:
The aim of the course is to introduce basic theories and methods in stochastic calculus for applications in stochastic control & optimization, financial mathematics and signal theory.

Intended learning outcomes:
To pass the course, the student should be able to do the following:

  • Be able to define and account for conditional expectation, filtrations and the martingale property in discrete and continuous time.
  • Account for the properties of the Brownian motion, with applications.
  • Define and account for Ito's stochastic integrals, Ito's lemma, Girsanov transform, the Martingale Representation Theorem.
  • Account for and determine strong and weak solutions of stochastic differential equations of Ito type (diffusion processes).
  • Account for and determine stochastic representations of solutions of parabolic partial differential equations (Kolmogorov's forward and backward equations, Feynman-Kac and Dynkin's formulas).

To receive the highest grade, the student should in addition be able to do the following:

  • Combine all the concepts and methods mentioned above in order to solve more complex problems.
Prerequisites:
SF 2940 Probability Theory or equivalent course.

Examination:

The written exam is on Thursday March 16, 2016, 14.00-19.00.

The written re-exam is on on Friday June 9, 2016, 14.00-19.00.

Grades are set according to the quality of the written exam. Grades are given in the range A-F, where A is the best and F means failed, and Fx. Fx means that you have the right to a complementary examination (to reach the grade E). The criteria for Fx is a grade F on the exam, and that an isolated part of the course can be identified where you have shown a particular lack of knowledge and that the examination after a complementary examination on this part can be given the grade E.
The exams are closed book with no aids allowed.

Students requiring assistance during the exams should contact KTH FUNKA in advance.

Homework:
There will be two homework assignments. Note that solving the homework assignments is not a requirement for passing the course. Correctly solved homework assignments handed in on time will result in that you will not have to solve parts of, or the whole of the first exercise on the exam in March and the re-exam in June, but after that you either have to re-do the homework or solve all exercises at the exam.

Learning units:
This course is structured in three learning units (learning cycles). Each unit consists of 4-5 lectures, 2 exercise sessions and 1 repetition session. Content of the units:

Learning Unit I: Conditional expectation, martingales and stochastic integrals in discrete time, Girsanov Theorem.

Learning Unit II: Martingales in continuous time, Brownian motion, Ito integral and Ito Lemma.

Learning Unit III: Martingale Representation Theorem, stochastic differentiable equations, Ito diffusions, Kolmogorov equations, Feynman-Kac formula.

Course literature:
Boualem Djehiche: Stochastic Calculus. An Introduction with Applications.
Available from "THS Kårbokhandel", Drottning Kristinas väg 15-19.

A list with further literature is found here.

Reading suggestions:
Please check out the reading suggestions here. They are updated after every lecture.

Preliminary plan (KS=Kevin Schnelli, MF=Martina Favero) 

Day Date Time Place Topic Lecturer
Tu 17/1 13-15 K2
Conditional expectation
KS
We 18/1 15-17 E2
Martingales in discrete time I
KS
Th 19/1 15-17 E2
Martingales in discrete time II
KS
Mo 23/1 10-12 B3
Stochastic integrals in discrete time, Girsanov I
KS
Tu 24/1 13-15 V3
Girsanov II. Martingales in continuous time I
KS
We 25/1 15-17 E2
Exercise session
MF
Th 26/1 15-17 E2
Martingales in continuous time and Brownian motion
KS
Mo 30/1 10-12 B2
Repetition learning unit I
KS
Tu 31/1 13-15 V3
Martingale property of Brownian motion
KS
We 1/2 15-17 E2
Exercise session
MF
Th 2/2 15-17 E2
Girsanov III, Ito integral I
KS
Mo 6/2 10-12 B1
Ito integral II
KS
Tu 7/2 13-15 E3
Ito integral III and Ito Lemma
KS
We 8/2 15-17 E2
Exercise session
MF
Tu 14/2 13-15 V3
Martingale representation theorem
KS
We 15/2 15-17 E2
Exercise session
MF
Mo 20/2 10-12 B1
Girsanov theorem IV, stochastic differential equations
KS
Tu 21/2 13-15 E3
Linear SDEs, strong solution to SDEs, repetition learning unit II
KS
We 22/2 15-17 E2
Exercise session, repetition learning unit II
MF
Mo 27/2 10-12 E3
Ito diffusions and their generators, Dynkin formula and Kolmogorov forward equation
KS
Tu 28/2 13-15 E3
Exercise session
MF
We 1/3 15-17 E2
Feynman-Kac formula, Kolmogorov backward equation
KS


To course web page

Published by: Kevin Schnelli
Updated: 02/03-2017