Stochastic integration 2015-2016
(code ST409018)

Announcements

  • There will be no lecture on May 5th, and no lecture on May 26th, but there will be an extra question session on June 2nd.
  • The exam will be on Monday, June 6th, from 9:00 to 12:00. You can bring your MTP and stochastic integration lecture notes to the exam, as well as your personal (hand-written) notes.

Contents

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, construction and properties of the stochastic integral, Itô's formula, Girsanov's theorem, stochastic differential equations and we will briefly explain their relevance for mathematical finance.

Prerequisites

Measure theory, stochastic processes at the level of the course Measure Theoretic Probability

Literature

Recommended background reading: I. Karatzas and S.E. Shreve, Brownian motions and stochastic calculus and D. Revuz and M. Yor, Continuous martingales and Brownian motion. The contents of the course are described in the (based on these books) lecture notes.

Companion course

Students are recommended to take also the course on Stochastic Processes, see the Spring Courses of the Dutch Master Program in Mathematics.

Follow up courses

A course that heavily relies on stochastic calculus is Interest rate models (the webpage is a bit outdated, but still fine for a first impression).

Lecturers

Sonja Cox (part II) and Peter Spreij (part I), assisted by Nicos Starreveld

Homework

Strict deadlines: the lecture after you have been given the assignment, although serious excuses will always be accepted. You are allowed to work in pairs (a pair means 2 persons, not 3 or more), in which case one set of solutions should be handed in.

Schedule

Spring semester: First lecture on Thursday 4 February 2016, 13:00-15:00. Lectures until 17 March in G2.04 (Science Park), from March 31 on in G0.10 but see also datanose.nl for up to date information. See the map of Science Park and the travel directions (in Dutch only). No lecture on March 3, March 24, May 5th and May 26. There is a lecture on April 28!.

Examination

The final grade is a combination of the results of the take home assignments and the oral exam (first part) and written exam (second part). To take the oral exam, you make an appointment for a date that suits your own agenda. If it happens that you'd like to postpone the appointment, just inform us that you want so. This is never a problem! The only important matter is that you take the exam, when you feel ready for it. What do you have to know? The theory, i.e. all important definitions and results (lemma's, theorems, etc.). Optional: you may prepare three theorems together with their proofs. You select your favorite ones! Criteria to consider: they should be interesting, non-trivial and explainable in a reasonably short time span. You will be asked to present one of them. Unavailable periods will appear here.

Registration

The UvA now wants all participants to be registered four weeks before the start of the course. If you missed this deadline you can use the late registration form. Note that a UvAnetID is required, so at least you have to be registered as a UvA student.


Programme

(regularly updated, Last modified: 01 April 2016 13:59)

1 Lecture: Sections 1 and 2.1 (very briefly).
Homework: Read the lecture notes, including the superficially treated Section 2.1 and make Exercises 1.4, 1.5, 1.8, 1.14.
2 Lecture: most of Section 2.2, Section 2.3: definitions, Lemma 2.15, proof of uniqueness of DM decomposition, Theorem 2.17 mentioned and Proposition 2.18.
Homework: Make yourself familiar with the contents of (Appendix) Sections B and D; just the big picture, no details. Make Exercises 2.1, 2.3 (optional), 2.5, 2.10.
3 Lecture: Remainder of Chapter 2 (proof of existence of DM decomposition, Proposition 2.18), introductory remarks on Chapter 3.
Homework: make Exercises 2.7, 2.8, 2.13, and (optional) 2.16.
4 Lecture: Chapters 3 and 4. plus initial notation from Chapter 5.
Homework: Exercises 3.3 (a,b,c), 3.9, 4.3
5 Lecture: Chapter 5 and quick introduction to Chapter 6
Homework: Exercises 4.10, 5.1, 5.2
6 Lecture: Most of Sections 6.1, 6.2, 6.3.
Homework: 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 6.1, 6.6, 6.8. (adjusted on March 17, 20:30)
7 Lecture: Sections 7.1, 7.2, 7.3
Homework: Make exercises 6.10, 7.1, 7.4, 7.5
8 Lecture: Section 7.4 and Chapter 8
Homework: 7.8 and 8.3. Also please have a look at exercises 7.6 and 8.1. (Sonja: I'm afraid that at the lecture I said 8.2 would be homework so if you solve that instead of 8.2 it is also OK but it is a more boring exercise in my opinion.)
9 (April 14) Lecture: Section 8.2 (cont'd), Sections 9.1 and 9.2
Homework: 8.5, 8.6, 9.3
10 (April 21) Lecture: Sections 9.3 and 9.4
Homework: 9.6, 9.12 (these are both relatively easy and you can score at most 8pts by solving them), 9.9 (I have not figured this one out yet myself, you can score at most 3pts by solving it -- of course your final grade will be min(10,#pts scored).)
11 (April 28) Lecture: Section 10.1
Homework (due May 12th!) 10.3, 10.4, 10.6 (read the proof of Propositions 10.3 and 10.4 first). You can also look at 10.8, which you can already solve but is due May 19th.
12 (May 12) Lecture: Section 10.2, p. 81.
Homework: 10.8, 10.12
13 (May 19) Lecture: Section 10.3, Chapter 11.
Homework: Option 1: 11.3 (note the conditions on b and sigma on p. 89) and 11.4, Option 2: 10.14 and 11.4. To be handed in on June 2nd.
14 (June 2nd) Lecture: Question session
Homework: no homework


To the Korteweg-de Vries Instituut voor Wiskunde or to the homepage of the master's programme in Stochastics and Financial Mathematics.