Stochastic Integration

8 EC

Semester 2, period 4, 5

5374STIN8Y

Owner Master Mathematics
Coordinator dr. A. Khedher
Part of Master Mathematics, Master Stochastics and Financial Mathematics, Master Mathematics, specialization Algebra and Geometry, year 1Master Mathematics, specialization Analysis and Dynamical Systems, year 1Master Mathematics, specialization Mathematical Physics, year 1Master Mathematics, specialization Stochastics , year 1Master Stochastics and Financial Mathematics, specialization Data Analysis and Statistics, year 1Master Stochastics and Financial Mathematics, specialization Financial Mathematics, year 1Master Stochastics and Financial Mathematics, specialisation Probability and Decision Making, year 1Master Mathematics, specialisation Discrete Mathematics and Quantum Information, year 1

Course manual 2022/2023

Course content

Stochastic calculus is an indispensable tool in modern financial mathematics. In this course we present this mathematical theory. We treat the following topics from martingale theory and stochastic calculus: martingales in discrete and continuous time, the Doob-Meyer decomposition, construction and properties of the stochastic integral, Itô's formula, (Brownian) martingale representation theorem, Girsanov's theorem, stochastic differential equations and we will briefly explain their relevance for mathematical finance.

Study materials

Literature

Syllabus

  • https://staff.fnwi.uva.nl/p.j.c.spreij/onderwijs/master/si.pdf

Objectives

  • can explain the theory and construction of stochastic integrals,
  • are able to apply the Itô formula,
  • can explain different solution concepts of SDEs,
  • know how to apply measure changes for continuous semimartingales (Girsanov's theorem) and are able to calculate the new drift,
  • are able to compare SDEs and PDEs, and can calculate the probabilistic representation of solutions to PDEs,
  • are able to solve problems, where knowledge of the above topics is essential.

Teaching methods

  • Lecture
  • Self-study
  • Homework assignments

Learning activities

Activity

Number of hours

Lectures

26

Self-study

224

Attendance

This programme does not have requirements concerning attendance (TER-B).

Assessment

Item and weight Details

Final grade

1 (100%)

Tentamen

Inspection of assessed work

The manner of inspection will be communicated via the lecturer's website.

Graded homework will be weekly returned.

Fraud and plagiarism

The 'Regulations governing fraud and plagiarism for UvA students' applies to this course. This will be monitored carefully. Upon suspicion of fraud or plagiarism the Examinations Board of the programme will be informed. For the 'Regulations governing fraud and plagiarism for UvA students' see: www.student.uva.nl

Course structure

Weeknummer Lectures Exercises
1 Chapter 1  Exercises 1.4-1.5
2 Sections 2.1 and 2.2 Make yourself familiar with the contents of (Appendix) Sections B and D; just the big picture, no details. Make Exercises 2.1, 2.3. 
3 Sections 2.3 and 2.4 Make Exercises 2.7,  2.16. 
4

Chapter 3 and Section 5.1  

 Make Exercises 3.3 (a,b,c), 3.9.
5 Section 5.2 and introduction to Chapter 6. Make Exercises  5.1, 5.2.
6 Sections 6.1 and 6.2 Read (optional, just needed in the proof of the Kunita-Watanabe inequality) Sections 6.3 and 6.9 of the MTP lecture notes; also look at the proof of this inequality. Make Exercises 6.1, 6.6.
7

Chapter 4 and Section 6.3

  
8

Sections 7.1 and 7.2

 Read the first example of a local martingale that is not a martingale 
9 sections 7.3 and 7.4 Read also the parts of section 8 that have been skipped, look at exercise 8.1 (it tells you why the ordinary Brownian filtration is not right-continuous) . To be prepared for next week, read the lecture notes on the Radon Nikodym theorem (mainly the introduction and the statement of the theorem).
10

Section 8

  
11

Sections 9.1 - 9.3

  
12

Section 9.4, Sections 10.1 and 10.2

 Read Proposition 10.3 and the second example of a local martingale that is not a martingale. 
13

Sections 10.3 and 11.1 

  

 

Timetable

The schedule for this course is published on DataNose.

Contact information

Coordinator

  • dr. A. Khedher

Staff

  • Asma Khedher