We will sometimes use a bit of measure theory, partly covered in the lecture notes. For basic knowledge (definitions and theorems) of sigma-algebras, random variables and integrals (expectation), you may look at the MTP lecture notes, sections 1.1, 1.2, 1.5, 3.1, 3.2, 4.1, 4.2, 4.4. Other concepts will be treated during the course.
November | Monday 13: | Tutorial 8 |
Wednesday 15: | Lecture 9 | |
Monday 20: | Lecture 10 | |
Wednesday 22: | Lecture 11 | |
Monday 27: | Tutorial 9 | |
Wednesday 29: | Tutorial 10 | |
December | Monday 04: | Tutorial 11 | Wednesday 06: | None | Monday 11: | Lecture 12 | Wednesday 13: | Tutorial 12 | Monday 18: | Lecture 13 | Wednesday 20: | Tutorial 13 |
(UPDATED SCHEDULE, )
1 |
Lecture: sections 1.1 (partly, until p.7 half way), a bit of A.5.
Exercise class: 1.1, 1.2(a), A.12. Homework: read what has been treated during class (even if it was superficially) and make Exercises 1.3, 1.4, A.13. See this too as experimenting a bit with what is doable (or not). |
2 |
Lecture: Review of the foundations of probability theory, conditional expectations in the finite and general case.
Exercise class: Read Proposition A.11, Section A.6 up to Definition A.12. Make Exercises A.1, and A.15 if time permits. Prove Proposition A.11(iii). Show that the conditional expectation of Definition A.11 is a.s. unique. Show also that $X+Y$ is a random variable, if $X$ and $Y$ are random variables. What about $XY$? Homework: None |
3 |
Lecture: Remainder of Section 1.1, Section 1.2.
Exercise class: Make Exercises 1.11, 1.12, 1.13(a,b). Homework: Make Exercise 1.5, 1.8, 1.9 and read the main results of Section A.4 (unless you already know this). |
4 |
Lecture: Sections 2.1, 2.2 until the first part of the proof of Theorem 2.5.
Exercise class: Make Exercises 2.1, 2.3, 2.4, 2.5. Homework: Make Exercises 2.2, 2.13 (needed for next time) and show that Equation (2.7) holds true. Make yourself familiar with the results of Section A.2 (and ask questions next time if necessary). |
5 |
Lecture: Some essentials of characteristic functions and Hilbert spaces, remainder of Section 2.1, starting with the second part of the proof of Theorem 2.5. Markov (Proposition 2.10) and martingale property of Brownian motion
Exercise class: Make Exercises 2.6, 2.8 and Exercise 6 of the additional exercises. Show also that $E\exp(uZ)=\exp(\frac{1}{2}u^2)$ if $Z$ is standard normal and $u$ is real, simply by integration. Homework: Make Exercise 2.9, Exercise 5 of the additional exercises, and then Exercise 2.11. |
6 |
Lecture: Section 2.3, the beginning of Section 3.1, and the backward heat equation.
Exercise class: Make Exercises 3.1, 3.2 (for this exercise you quickly glance at the proof of Proposition 3.2, in particular Eq (3.4) and what is around it), 3.4. Homework: Make Exercises 2.10(b), 2.14. |
7 |
Lecture: Remainder of Section 3.1 (Proposition 3.2 + proof, mentioning of Thm 3.3) and Section 4.1 up to Proposition 4.2.
Exercise class: none Homework: Make Exercises 3.5, 3.6 and 3.9. |
8 |
Lecture: Section 4.1 from Corollary 4.3, Sections 5.1, 5.2.
Exercise class: Make Exercises 4.5, 4.7, 4.8, and 4.1 if time permits. Homework: Make Exercises 4.3, 4.6, 4.9, 5.1. |
9 |
Lecture: Section 5.3, Section 6.1 up to Proposition 6.1.
Exercise class: Make Exercises 5.3, 5.4(c,e) [the answer to (e) should be familiar to you], 6.1. Homework: Make Exercises 5.2, 5.4(a,b), 6.2. |
10 |
Lecture: Section 6.1 from Proposition 6.1, proof of Proposition 6.5 only for simple processes (planned recap of Section A.6 skipped).
Exercise class: Make Exercises 6.6, 6.7 and 6.12. Homework: Make Exercises 6.4, 6.5 and 6.8 (think of using quadratic variation). |
11 |
Lecture: Section 6.2.
Exercise class: Make Exercises 6.9, 6.13 and 6.16. Homework: Make Exercises 6.10, 6.11 and 6.19. |
12 |
Lecture: Sections 6.3, 7.1, Section 7.2 up to Proposition 7.4 / Corollary 7.5.
Exercise class: Make Exercises 4.12, 6.14, 7.5. Homework: Make Exercises 6.15 and 7.4. |
13 |
Lecture: Section 7.2 from Corollary 7.5, Sections 7.3, 7.4.
Exercise class: Make exercises 7.2, 7.6 (if enough time), 7.7 (don't give an explicit expression (unless you really want), but express the price of the straddle in terms of the prices of the constituents of the constant portfolio), 7.8, 7.11. Homework: None. |