Stochastic integration 2017-2018
(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 (2010 version)

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A course that heavily relies on stochastic calculus is Interest rate models (the webpage is a bit outdated, but still fine for a first impression).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

Asma Khedher (first half) and Peter Spreij (second half), assistance by Madelon de Kemp.

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, 15:00-16:45, first lecture on Thursday 15 February 2018 (we skip the first week!). For up to date information on the lecture rooms (mainly, but not always, B0.207), see datanose.nl. See also the map of Science Park and the travel directions. No lecture on February 8, other changes in the schedule will appear here or announced otherwise.

Examination

The final grade is a combination of the results of the take home assignments and the written or oral exam (first part, to be decided) and oral exam (second part).

To take the oral exam for the second part, you make an appointment with Peter 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 favourite 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 5-8 June, 18-21 June, 26-27 June, 29 June - 3 July; more will be announced here too. Look at the schedule that has been prepared after the final lecture (with later modifications on request as well). If you don't appear on the list, send me a mail. The reserved dates are 11 (afternoon), 13, 14, 22 June 2018. An extra date will be 25 June 2018, as the 22nd is already very full. Note that it is always possible (even at the last moment) to move the exam to a later date, if you feel that this would be beneficial for you. The exams (about 30 minutes, although 45 minutes have been allotted for each of you) will take place in Peter's office, room F3.35. The office is across the street in the NIKHEF building, Science Park 105, not in the main building, Science Park 904.

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.


Updated 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, 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 2.16 (skip the optional part).
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 up to Proposition 6.10.
Homework (updated on 5 April 2018): Make Exercises 6.5, 6.10, 6.13. (It is not necessary to make the distinction between bounded processes and the general case (as stated in the exercise) to give the proof. Jus rely on the relevant results in Chapter 5.)
8 Lecture: Sections 6.2 (from Theorem 6.11), 6.3, 7.1, 7.2 (perhaps).
Homework: Make Exercises 6.7, 6.9, 6.11, 6.14.
9 Lecture: Sections 7.2, 7.3, 7.4 (except BDG-inequality, you may read this yourself), 8.1 and the beginning of Section 8.2.
Homework: Read (just read) the first example of a local martingale that is not a martingale and make Exercises 7.1, 7.4, 7.5, 7.7 (restrict yourself to $f$ twice continuously differentiable in both variables and depart from the formula in Remark 7.12, then there will not be much left to do). Look (just look) also 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) if you are not familiar with it.
10 Lecture: Sections 8.2, 9.1, 9.2.
Homework: Read also the parts that I skipped (like Corollary 8.8 and Proposition 9.8). Make Exercises 8.2, 8.3 (only for $t < T$), 9.4 (only prove the statement "If $XZ$ is a local martingale under $\mathbb{P}$ then $X$ is a local martingale under $\mathbb{Q}$", and 9.6. [If this all turns out too much work, you may drop Exercise 9.6.]
11 Lecture: Most of Sections 9.3, 9.4, Introduction to Sections 10 and 10.1.
Homework: Make Exercises 9.7 (Message of Monday 14 May: there is a misprint in the exercise, $M$ should be $M^{\mathbb{Q}}$ as in the proof of Theorem 9.9.), 9.8, 9.9, 9.12.
12 Lecture: Most of Sections 10.1 and 10.2.
Homework: Make Exercises 10.3, 10.4, 10.7, 10.18 (in part (a) you have to show that the iteration results in the formula for $X^n_t$); read Proposition 10.3 and the second example of a local martingale that is not a martingale.
13 Lecture: Sections 10.3, 11.1 (most of it).
Homework: 11.1, 11.3 (optional), 11.4.


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