Below you find the theorems of which you are supposed to be able to
reproduce statements and proofs on the exam. The theory part of the exam
(20 of the 40 points) tests your knowledge of these.
1
Martingales
You are supposed to know all the definitions. From the following theorems
you are supposed to be able to reproduce the statement and proof. The
numbers refer to the lecture notes on martingales in discrete time.
1. Section 2: Theorem 2.2, 2.3.
2. Section 4: Theorem 4.1, 4.2, 4.3, 4.4
3. Convergence theorems: if I ask “state and prove the martingale convergence theorem”, then you can choose between the L2 -version (Theorem
5.3) in which case you are also asked to state and prove the KolmogorovDoob inequality (Theorem 5.1). Alternatively, you can choose to give
the up-crossing proof (Theorem 6.1), in which case you also have to
prove the up-crossing inequality (lemma 6.1, lemma 6.2).
4. You have to be able to provide at least one example of martingale
convergence (section 8).
2
Brownian Motion
You are supposed to know all the definitions. From the following theorems
you are supposed to be able to reproduce the statement and proof. The
numbers refer to the lecture notes on Brownian motion.
1. Section 1: Theorem 1.1, Proposition 1.3, Theorem 1.2, Theorem 1.3,
Theorem 1.5, proposition 1.5, Lemma 1.1, Theorem 1.6.
2. Section 2: Theorem 2.2, Proposition 2.3, lemma 2.1, Theorem 2.4
1