| 1a |
Class: Random walk, Central limit theorem, Brownian motion, martingale, quadratic variation; from Shreve, most (but not all!) of sections 3.1-3.4. |
| 1b |
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
TA session: Exercises 3,4, 3.6, 4.2 and 4.4 if time permits Homework: Exercises 3.1, 3.2, 4.1; read Section 3.2.7, a motivation for the use of Brownian motion in finance (but you have to know the Binomial model for stock prices...). |
| 2a |
Class: Itô integral continued, $\int_0^TW(t)\,\mathrm{d}W(t)$, Itô-formula for Brownian motion and Itô processes, deterministic integrands; from Shreve, theory of Section 4.4.6 |
| 2b |
Class: multivariate Brownian motion, product rule, Levy's characterization,
Change of measure, Girsanov's theorem; from Shreve, Sections 4.6.1, parts of 4.6.2 and 4.6.3, 5.2.1
TA session: Exercises 4.5, 4.6, 4.8, 5.1 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, 5.2 |
| 3a |
Class: Girsanov and risk neutral pricing in finance, martingale representation theorem, start with Poisson processes; from Shreve, Sections 5.2.2-5.2.4, 5.3, 11.2 |
| 3b |
Class: Compound Poisson process, integrals
w.r.t. jump processes; from Shreve, Section 11.3, 11.4
TA session: Exercises 5.5, 5.8, 5.11 (if there is enough time), Example 11.4.6 in detail. Extra: Given a compound Poisson process (notation as in Shreve), find a left continuous process $Y$ such that $Q(t)=\int_0^t Y(s)\,\mathrm{d}N(s)$. Show (argue) that $\mathbb{E}\int_0^t Y(s)\,\mathrm{d}s=\beta t$ and relate this to Theorem 11.3.1. Homework: Read Section 5.4.1 and (if you like) more on derivative pricing in Section 5.4 to get an idea why the change of measure is useful and necessary in finance, have a look at the Summary (Section 5.7) and the Notes (Section 5.8) as well. Make Exercise 8.3(a,b,c) 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 $t < T$ only). Don't get confused by the notation $X\cdot W$. It means stochastic integral, not a product (this notation is not uncommon, by the way, see also Williams for a discrete time version). Further: Exercises 11.1 (first (ii), than (i)), 11.2, 11.3 |