Stochastic integration 2018-2019
(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

In the past students were 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 (2018). This course has been replaced with a new one under the same name, but with different topics.

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A course that heavily relies on stochastic calculus is Interest rate models (the webpage is a bit outdated, but still fine for a first impression).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

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, 09:00-10:45, first lecture on Thursday 7 February 2019. Tutorials biweekly after the lectures. For up to date information on the lecture rooms, see datanose.nl. See also the map of Science Park and the travel directions. Changes in the schedule: no lecture and exercises on April 18, 2019.

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). Note, added on 29 May 2019: the homework results count as a bonus for 30% of the final grade. The partial oral exams have equal weight.

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. Make an appointment with Peter for a date that suits you well. Impossible dates are May 22-31 (perhaps with an exception on May 29), June 3, 6, 10-24, 26, July 23 - August 20.

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.


1st half, old programme

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. Just rely on the relevant results in Chapter 5.)

2nd half, Updated programme

(regularly updated!, )

8 Lecture: Sections 6.2 (from Theorem 6.11), 6.3, 7.1, 7.2.
Homework: Make Exercises 6.7, 6.9, 6.12, 6.14. (changed on 9 April 2018)
9 Lecture: Sections 7.2, 7.3, 7.4, 8.1 and the beginning of Section 8.2.
Homework: Read (just read, you can also do this a week later) 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 (most of it), 9.1, 9.2 (most of it), 9.3 up to Corollary 9.13.
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.
11 Lecture: Most of Sections 9.3, 9.4, Section 10.1 at top speed up to Theorem 10.2.
Homework: Make Exercises 9.7, 9.9, 9.12, 10.3.
12 Lecture: Remainder of Section 10.1, most of Section 10.2 and a beginning of Section 10.3.
Homework: Make Exercises 10.4, 10.7, 10.9 (you may drop this exercise if it is too much work), 10.18 (in part (a) you have to show that the iteration results in the formula for $X^n_t$); read the second example of a local martingale that is not a martingale.
13 Lecture: Sections 10.3, 11.1 (most of it).
Homework: 10.14, 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.