Measure theory and stochastic processes
2016-2017 (TI083)
Tinbergen Institute

Aim

To make students familiar with the mathematical fundamentals of, measure theory, stochastic processes and stochastic integrals. This is a crash course, highlighting the main principles, not an in depth treatment of the theory.

Contents

Sigma-algebras, measure, integration w.r.t. a measure, limit theorems, product measure and integration, change of measure, conditional expectation; Heuristic construction of Brownian motion, martingale property and quadratic variation; construction of the Itô integral, fundamental properties (Itô isometry); Itô rule (in one and more dimensions), stochastic product rule, Lévy's characterization of Brownian motion; absolutely continuous change of measure, Girsanov's theorem, martingale representation theorem; and if time permits stochastic differential equations, diffusions and partial differential equations, Feynman-Kaç formula.

Literature

The course is mainly based on Steve Shreve (2004), Stochastic Calculus for Finance II, Continuous-Time Models, Springer (We will use this book, buy it!). There are many other texts on measure theory, stochastic processes and stochastic integration. For free are available the lecture notes Measure theoretic probability, An Introduction to Stochastic Processes in Continuous Time and Stochastic integration. These notes are used for courses in the master programme for mathematics students Stochastics and Financial Mathematics, but will not be used for the present course.

Examination

We will follow the usual conventions for TI core courses, i.e. there will be a written closed book exam. The exercises at the exam will be at the level of the homework and tutorial sessions, but may also contain some theory. You don't have to know all proofs by heart, but at least the gist of them. Important theorems and definitions you are required to know. The written exam is on October 17, 2016, 14:30-17:30 in room 1.01.

People

Lectures by Peter Spreij, teaching assistance by Jun Huang.

Schedule

Fall semester, 1st half. Here is the list of lectures. Changes are not expected anymore (I have thought this before...), but one never knows.
  1. Fri 2 September, 15:00 - 17:45
  2. Wed 7 September, 10:00 - 12:45
  3. Wed 7 September, 13:30 - 16:15
  4. Wed 21 September, 13:30 - 16:15
  5. Wed 28 September, 13:30 - 16:15
  6. Wed 5 October, 13:30 - 16:15
  7. Wed 12 October, 13:30 - 16:15
Tutorials usually on Fridays, 11:00-12:00, first tutorial on Monday 5 September; look out for changes.

Location

Tinbergen Institute Amsterdam, Gustav Mahlerplein 117, 1082 MS Amsterdam

Programme
(please, look out for updates; )

1
Class: Most of Sections 1.1, 1.2, 1.3
Homework : Read before the second class in the book the sections that I have treated in the classroom, including those pieces that have been skipped. In particular pay attention to the Lebesgue integrals following Definition 1.3.7 and observe the similarities and differences with expectations. Take notice of (the gist of what is written in) Sections A.1 and read also section A.2.
Tutorial: Exercise 1.5 and one or two exercises from the additional exercises (probably exercises 2 and 3)
Homework: Read the Sections that have been treated in the second lecture, also the parts and examples that have been skipped. Read also pages 49, 50 as an introduction to the next class. Make Exercises 1.1, 1.2, 1.3 of Section 1.9.
2
Class: Most of Sections 1.4, 1.5, 1.6 and Appendix B
Tutorial: 1.8, 1.15, additional Exercises 5, 7 (if there is enough time, otherwise drop one exercise)
Homework: 1.9, 1.10, 1.11, additional Exercise 8.
3
Class: Most of Sections 2.2, 2.3 (Definitions 2.3.5 and 2.3.6 will be treated next time, perhaps).
Tutorial: Exercises 2.2, 2.10 and 2.4 if there is time left.
Homework: Make Exercises 2.1, 2.6, 2.7, 2.9 and read the Summary Sections 1.7 and 2.4. Also read the examples in Shreve, and pay special attention to the first three pages of Section 2.3, but ignore Equations (2.3.1) - (2.3.3) and the surrounding text.
4
Class: Sections Sections 3.2 - 3.4
Tutorial: Exercises 3.1, 3.4, 3.6
Homework: Make Exercise 3.2, 3.5 and compute quadratic variation of $W(t)+at$ over $[0,T]$; read Section 3.5 and the relevant part of Section 3.8.
5
Class: Sections 4.2, 4.3, 4.4.1 and 4.4.2
Tutorial: Exercises 4.1, 4.4
Homework: Exercises 4.2, 4.3, 4.5 (added on September 28)
6
Class: Remainder of Section 4.4, most of Sections 4.6 (of 4.6.3 the one-dimensional case only) and Sections 6.2, 6.3, 6.4.
Tutorial: Exercises 4.9, 4.14, 6.1 (changed on October 5).
Homework: Carefully read Sections 4.4, 4.6, in particular the examples. Make Exercises 4.7, 6.8, 6.9 (changed on October 5).
7
Class: Sections 5.2, 5.3.
Tutorial: Exercises 5.1, 5.8
Homework: Make Exercise 5.5 (more follows!) and the following exercise: Let $T>0$ and $M_t=\mathbb{E}[\int_0^T W(s)\,\mathrm{d}s\,|\mathcal{F}(t)]$ for $t\in [0,T]$. Here $W$ is a Brownian motion and $\mathcal{F}(t)=\sigma(W(s),s\in [0,t])$.
(a) Show that the process $M$ is a martingale.
(b) Find the $\Gamma(s)$ of Theorem 5.3.1 for $s\in [0,T]$.




Links

Korteweg-de Vries Institute for Mathematics
Master Stochastics and Financial Mathematics