Coursework 1 - MATH3734 /5734M
This piece of coursework is worth 5% of the inal module mark and it is marked out of 20. You must write down your answers on a piece of paper (handwritten or typeset in Latex) and submit them via Minerva. Deadline for submission is 28 March at 12 pm (noon).
IMPORTANT NOTE: This is a take-home coursework and you have 1 week to complete it. This means that you have access to lecture notes and solved exercises. If you use results from the lecture notes or the exercises state exactly which Theorem/Lemma/Proposition/Exercise/Remark you are using.
Problems
Problem 1 (6 marks). Let Ω be a set. A collection G of subsets of Ω is called an algebra on Ω if the following are satisied:
1. ∅ ∈ G,
2. if A ∈ G then Ac ∈ G,
3. if A, B ∈ G then A ∪ B ∈ G.
Let Ω = N be the set of all natural numbers (i.e. {1, 2, 3, ...}). Let G denote the collection of subsets A of Ω with the following property: either card(A) < ∞ or card(Ac ) < ∞ (where by card(B) we denote the cardinality of a set B, that is, the number of elements that it contains).
(i) Show that G is an algebra.
(ii) Show that G is not a σ-algebra.
Problem 2 (6 marks). Let (Ω , F, P) be a probability space. Let (W (t))t2[0 ;1) be a one dimensional Wiener process process on (Ω , F, P). For n ∈ N, consider the random variables Xn = W ((n + 1)2 ) - W (n2 ).
1. Explain why (Xn)∞n=1 are independent.
2. For n ∈ N, ind the distribution of Xn.
3. Use Markov’s inequality to show that
P (|Xn | > n√1 + 2n, ≤ n2/1
4. Use the Borel-Cantelli lemma to show that with probability one, there exist only initely many n ∈ N such that |Xn | > n√1 + 2n.
Problem 3 (8 marks). Let (Ω , F, P) be a probability space, T = 10, and let and F = (Ft )t2[0,T] be a iltration of sub-σ-algebras of F. Consider a one dimensional F-Wiener process (W (t))t2[0,T] on (Ω , F, P).
1. Consider three random variables Yi , for i = 1, 2, 3 which are given by
Yi = i · 1fW (i)>0g ,
that is, Yi = i if W (i) > 0, and Yi = 0 otherwise.
(a) Explain why Yi is Fi-measurable for each i = 1, 2, 3.
(b) Consider the process
Z(t) = Y11[1 ,2) (t) + Y21[2 ,3) (t) + Y31[3 ,4) (t).
Show that Z ∈ HT(step) .
(c) Write down the deinition of the stochastic integral
l010 Z(s) dW (s).
(d) Find the expectation and the variance of the stochastic integral from (c).