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 $s\in [0,T]$. |