Stochastic integration 2008-2009

Contents

Stochastic calculus is an indispensable tool in modern financial mathematics. In this course we present this mathematical theory and apply it to the problem of pricing and hedging of financial derivatives. We treat the following topics from martingale theory and stochastic calculus: martingales in discrete and continuous time, construction and properties of the stochastic integral, Ito's formula, Girsanov's theorem, stochastic differential equations. As an application, we explain how stochastic differential equations are typically used to model financial markets and we discuss the problem of the pricing of derivatives such as stock options.

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 lecture notes (based on these books).

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

Courses that heavily rely on stochastic calculus are Financial Stochastics and Control of Stochastic Systems in Continuous Time.

Lecturer

P.J.C. Spreij

Homework

Strict deadlines: the lecture after you have been given the assignment, although serious excuses will always be accepted.

Schedule

Spring semester: usually Thursdays, 15:00-17:00, Room P.015B. First lecture on 5 February 2009, 15:00-18:00. No classes on April 30, May 7. On April 23 and May 14 (last lecture), lectures from 14:00 to 17:00.

Examination

The final grade is a combination of the results of the take home assignments and the oral exam. To take the oral exam, you make an appointment for a date that suits your own agenda (and mine as well). If it happens that you'd like to postpone the appointment, just inform me 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. To plan your own agenda, please note that I am not always in Amsterdam. As far as is known, I am NOT available during the periods June 9 - 20, June 29 - July 5, July 15 - 17, July 27 - August 21. However, changes may occur, so always check this before you want to set up a meeting.


Programme

Please, regularly check the programme below.
It is not definitive and adjustments will be made throughout the course

1 Lecture notes: Sections 1, 2.1
Homework: Exercises 1.1, 1.4, 1.5, 1.14
2 Lecture notes: Sections 2.2, 2.3 (Lemma 2.14, uniqueness of DM decomposition, Proposition 2.17 briefly)
Homework: make Exercises 2.1, 2.3, 2.5, 2.9, read the proof of Proposition 2.17
3 Lecture notes: Section 2.3 (existence part of the proof of Theorem 2.14, Theorem 2.15 mentioned)
Homework: make Exercises 2.6, 2.7, 2.12, 2.15 (new exercise, see last version of the lecture notes), have a look at Appendix C
4 Lecture notes: most of Sections 3 and 4
Homework: Exercises 3.3, 3.6, 3.9, 3.10 (second part only)
5 Lecture notes: Section 5 and the beginning of section 6
Homework: Exercises 4.2, 4.3, 5.1, 5.2
6 Lecture notes: Sections 6.1, 6.2 up to Theorem 6.11, without the proof of Lemma 6.9.
Homework: Read (optional, just needed in the proof of the K-W inequality) sections 3 and 6 of "The Radon-Nikodym theorem" (part of MTP lecture notes); make 6.1, 6.6, 6.9. Strict deadline: 26 March, 17:00!
7 Lecture notes: Sections 6.2 (remaining parts), 6.3, 7.1
Homework: Make exercises 6.10, 6.11, 7.1
8 Lecture notes: Sections 7.2, 7.3, 7.4, 8.1
Homework: Read the first example of a local martingale that is not a martingale and make Exercises 7.1 (only the last question, you may assume the other results to be known), 7.4, 7.6 (you only have to deduce this from (7.8)). Strict deadline: 9 April, 17:00!
9 Lecture notes: Section 8 until Theorem 8.6, a quick review of Propositions 9.3, 9.4, 9.5
Homework: Make exercises 8.1, 8.2, 8.3
10 Lecture notes: Lemma 9.2, Proposition 9.7 (partly), most of Section 9.3, highlights of Section 9.4
Homework: Make exercises 9.4, 9.5, 9.8
11 Lecture notes: Sections 10.1 (Proposition 10.3 only mentioned, Theorems 10.4 and 10.5 skipped), 10.2 (proof of Proposition 10.7 only sketched)
Homework: Make exercises from 10.4, 10.9, 10.16. Strict deadline for all remaining exercises up to and including those of week 11: 14 May, 17:00!
12 Lecture notes: sections 10.3 and 11
Homework: Make exercises 11.1, 11.2, 11.3 and read the second example of a local martingale that is not a martingale. Strict deadline: 25 May, 17:00!


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