| 1 |
Class: Random walk, Central limit theorem, Brownian motion,
martingale, quadratic variation; from Shreve, most (but not all!) of sections 3.1-3.4.
Homework: Exercises 3.1, 3.2, 3.4 (read also the additional text in Section 3.4.2), 3.5. |
| 2 |
Class: Integral of simple functions, construction of stochastic
integral w.r.t. Brownian motion, properties (isometry), general
integrands; from Shreve, most of sections 4.1 - 4.3
Homework: Exercises 4.1, 4.2 |
| 3 |
Class: Itô-formula, bivariate extension, product rule, Lévy's
characterization; from Shreve, theory of Section 4.4, parts of Section 4.6
Homework: Understand the examples in Section 4.4, pay attention to the parts of Section 4.6 that I skipped and make exercises 4.7, 4.14 |
| 4 |
Class: Change of measure, Girsanov's theorem, Martingale
representation theorem, relevance
for risk neutral pricing; from Shreve, Sections 5.2.1-5.2.4, 5.3
Homework: Read Section 5.4.1 and (if you like) more on derivative pricing, have a look at the Summary (Section 5.7) and the Notes (Section 5.8) as well. Make Exercises 5.2, 5.5 and Exercise 8.3 from the lecture notes of another course in which you have to find the process Γ of Theorem 5.3.1 for the martingales in the exercise. Don't get confused by the notation X • W. It means stochastic integral, not a product (this notation is not uncommon, by the way, see also Williams for a discrete time version). |
| 5 |
Class: Stochastic differential equations, connections to Partial
differential equations, Feynman-Kac formula, stopping times; from Shreve, Sections 6.2 - 6.4, 8.2 and Definition 8.3.1
Homework: Read Sections 6.7, 6.8 (as far as applicable), 6.5 ; make Exercises 6.1, 6.8, 6.9; extra question: suppose that X is an adapted continuous process. Define τ as the first moment t that X(t) is at least equal to some level m, τ = min{t≥ 0: X(t)≥ m}. Give a simple argument that shows that τ is a stopping time. (Distinguish between X(0)≥ m and X(0) < m.) |
| 6 |
Class: (Compound) Poisson process, integrals
w.r.t. jump processes; from Shreve, Chapter
11 up to Theorem 11.4.5
Homework: Exercises 11.1(ii), 11.2, 11.3 |
| 7 |
Class: quadratic variation, Ito-formula for processes with jumps, change of measure
for (compound) Poisson process; from Shreve, Chapter 11, Theorem 11.4.5 - Theorem 11.6.5 with an informal discussion of Theorems 11.6.7 and 11.6.9
Homework: Exercises 11.4, 11.5, 11.7, Watanabe's characterization of a Poisson process and only read 11.6 in order to know the result. |