Stochastic integration 2011-2012
(code ST409018)

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, Itô'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 (which are based on these books, updated version composed during this spring).

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.

Lecturers

Peter Spreij, assisted by Naser Asghari.

Homework

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

Schedule

Spring semester: Thursdays, 11:00-13:00, Room B0.209 (Science Park); see the map of Science Park and the travel directions. First lecture on 9 February 2012 (starting at 10:00), March 1 class: 09:00-11:00 taught by Neil Walton, no class on March 29.

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. 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. I'll then ask you to present two of them. I am unavailable in the periods 17-22 May, 6-14 June, 29 June - 4 July and more....


Programme

(last modified: )

1 Lecture: Sections 1 and 2.1.
Homework: Exercises 1.4, 1.5, 1.8, 1.14
2 Lecture: Section 2.2, Section 2.3: definitions, Lemma 2.15, proof of uniqueness of DM decomposition, Theorem 2.17 and Proposition 2.18 mentioned.
Homework: Make yourself familiar with the contents of (Appendix) Sections A and C (just the big picture, no details). Make Exercises 2.1 (ignore the hint, it is not useful), 2.3 (optional!), 2.5, 2.9.
3 Lecture: Remainder of Chapter 2 (proof of existence of DM decomposition, Proposition 2.18), introductory remarks on Chapter 3.
Homework: make Exercises 2.6, 2.7, 2.12, and (optional) 2.15
4 Lecture: Chapter 3 and a bit of Chapter 4
Homework: Exercises 3.3 (total variation is the same as first order variation), 3.6 (optional,this is a `stand alone' exercise), 3.9, 3.10
5 Lecture: Chapter 4 and Chapter 5
Homework: Exercises 4.1, 4.3, 5.1, 5.2
6 Lecture: Most of Sections 6.1, 6.2
Homework: Read (optional, just needed in the proof of the Kunita-Watanabe inequality) Sections 6.3 and 6.6 of the MTP lecture notes; also look at the proof of this inequality. Make 6.1, 6.6, 6.9.
7 Lecture: Sections 6.2 (remaining parts), 6.3, 7.1
Homework: Make exercises 6.8, 6.10, 6.11
8 Lecture: Sections 7.2, 7.3, 7.4
Homework: (plan) Read the first example of a local martingale that is not a martingale and make Exercises 7.1, 7.4, 7.5, 7.6 (restrict yourself to f twice continuously differentiable in both variables).
9 Lecture: Section 8
Homework: Read also the parts of section 8 that I skipped, look at exercise 8.1 (it tells you why the ordinary Brownian filtration is not right-continuous) and make exercises 8.2, 8.3, 8.6.
10 Lecture: Sections 9.1 - 9.3, quick mentioning of section 9.4
Homework: Make exercises 9.4, 9.6, 9.8, 9.9 (assume that M is continuous!!)
11 Lecture: Sections 9.4, 10.1 (up to Theorem 10.2)
Homework: Read the end of the proof of Theorem 10.2 and make exercises 9.11, 9.12, 10.3, 10.7.
12 Lecture: Sections 10.2, beginning of 10.3
Homework: Make exercises 10.4, 10.12, 10.16 and read the second example of a local martingale that is not a martingale.
13 Lecture: Sections 10.3, 11.1 (without the proof of Theorem 11.2)
Homework: 11.1, 11.3, 11.4



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