Stochastic integration 2016-2017
(code 5374STIN8Y)

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, 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.

Aims

At the end of the course, students
  • 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.

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A course that heavily relies on stochastic calculus is Interest rate models (the webpage is a bit outdated, but still fine for a first impression).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

Peter Spreij and Sonja Cox (March 16, April 6), assistance by Mike Derksen

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: Thursdays on 16:15-18:00 (note the odd hours), first lecture on Thursday 9 February 2017. No lecture on March 2, but on Monday February 27 instead, from 09:00 to 11:00. Lectures in many varying(!) lecture rooms, for up to date information see datanose.nl. See also the map of Science Park and the travel directions (in Dutch only). Further, no lectures on March 30, April 27 and May 25.

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 are: June 19-23 (although perhaps the afternoon of June 23 could be possible), and July 11 - August 13.

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, )

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), perhaps introductory remarks on Chapter 3.
Homework: make Exercises 2.7, 2.8, 2.13, and (optional) 2.16.
4 Lecture: Chapters 3 and 4 (rather briefly).
Homework: Exercises 3.3 (a,b,c), 3.9, 4.3
5 Lecture: Chapter 5 and perhaps a quick introduction to Chapter 6
Homework: Exercises 4.10, 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 6.9 of the MTP lecture notes; also look at the proof of this inequality. Make 6.1, 6.6, 6.8.
7 Lecture: Sections 6.2 (from Theorem 6.11), 6.3, 7.1.
Homework (updated on 24 March 2017): Make exercises 6.9, 6.10, 6.13.
8 Lecture: Sections 7.2, 7.3, 7.4
Homework: (will be changed?) 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 and depart from the formula in Remark 7.12).
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 (only for t < T), 8.6 (optional). 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 Lecture: Sections 9.1 - 9.3.
Homework: Make exercises 9.4, 9.6, 9.8, 9.9.
11 Lecture: Section 9.4, Sections 10 and 10.1 up to Theorem 10.2.
Homework: Make exercises 9.5, 9.12, 10.3, 10.9 .
12 Lecture: Most of Sections 10.1 (remainder) and 10.2
Homework: Make exercises 10.4, 10.7, 10.18; read Proposition 10.3 and 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


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