| 1 |
Class: Random walk, Central limit theorem, Brownian motion,
martingale, quadratic variation; from Shreve, most (but not all!) of Sections 3.1-3.4.1.
TA session: Exercises 3.5, 3.6; as a preparation you have a look at Section 3.5. Homework: Exercises 3.1, 3.2. |
| 2 |
Class: Quadratic variation of Brownian motion, stochastic integral of simple functions; from Shreve, most of Section 4.2.
TA session: Exercises 3,4, 4.2. Homework: Exercises 4.1, 4.3. |
| 3 |
Class: Itô integral continued, $\int_0^TW(t)\,\mathrm{d}W(t)$, Itô-formula for Brownian motion and Itô processes, from Shreve, Sections 4.3. 4.4.
TA session: Exercises 4,4, 4.5. Homework: Exercises 4.6, 4.7. |
| 4 |
Class: Itô integrals with deterministic integrands; multivariate Brownian motion, product rule, Levy's characterization;
from Shreve, remainder of Section 4.4, Sections 4.6.1, most of Sections 4.6.2 and 4.6.3.
TA session: Exercises 4.10 (lengthy story, but there is not much work), 4.13, 4.18. Homework: Understand the examples in Section 4.4, pay attention to the parts of Section 4.6 that I skipped, and make Exercises 4.8, 4.14, 4.15. |
| 5 |
Class: Levy's characterization, change of measure, Girsanov's theorem; risk neutral pricing in finance; from Shreve, Sections 5.2.1-5.2.4, perhaps 5.3.
TA session: Exercises 4.9 (recall Exercise 3.5), 5.3. Homework: Read Section 5.2.5, make Exercises 5.1, 5.2. |
| 6 |
Class: martingale representation theorem, start with Poisson processes; from Shreve, Sections 5.3, 11.2.
TA session: Exercises 5.5. 5.12 (for which you have to pick up the essentials of Sections 5.4.1 and 5.4.2, they are multivariate versions of what you already know). Homework: Exercises 5.8, 5.11. |
| 7 |
Class: Compound Poisson process, integrals
w.r.t. jump processes; from Shreve, Section 11.3, 11.4 and some highlights from Section 11.5.
TA session: Exercises 11.3, 11.4; Exercise 8.3(b) from the lecture notes of another course in which you have to find the process $\Gamma$ of Theorem 5.3.1 for the martingales in the exercise (do this for $t \leq T$ only). Don't get confused by the notation $X\cdot W$. It means the stochastic integral $\int X(s)\,\mathrm{d}W(s)$, not a product (this notation is not uncommon, by the way, see also Williams for a discrete time version). Homework: Exercise 8.3(b) from the lecture notes as above. Further: Exercises 11.1 (first (ii), than (i)), 11.2. |