1 Homework 1
to be handed in: Febr. 24, 2016 Exercise 1 Problem 2.1 from BN (Background Notes).
Exercise 2 Problem 4.1A from BN. You may assume the result of Problem 4.1.
Excercise 3 Consider the construction of the process N = (Nt)t≥0 on p.4 of the LN. So, Nt(ω) < ∞ for all ω ∈ Ω, t ≥ 0. Then
N : (Ω, F , P) → (Z[0,∞)+ , (2Z+)[0,∞)),
where the σ-algebra (2Z+)[0,∞) is the smallest σ-algebra that makes the projection map Πt: Z[0,∞)+ → Z+
(2Z+)[0,∞)/2Z+-measurable for any t ≥ 0. This is the map Πt(x) = xt, for x ∈ Z[0,∞)+ . I.o.w.:
Πt: (Z[0,∞)+ , (2Z+)[0,∞)) → (Z+, 2Z+).
Note that 2Z+ is the collection of all subsets of Z+ (power set). Further, according to our discussion, (2Z+)[0,∞) consists of σ-cylinders.
The above implies that the map ω → Nt(ω) is F /2Z+-measurable, as a composition of measurable maps.
a) Specify the simplest possible π-system for both 2Z+ and the σ-algebra (2Z+)[0,∞).
b) Consider now the process Ns= (Nt)t≤s, for some s ≥ 0. This is the process N restricted to index set [0, s]. Specify the smallest σ-algebra that makes the projection map Πst : Z[0,s)+ → Z+ measurable for all t ∈ [0, s]. Give the simplest possible π-system for this σ-algebra.
c) Determine a π-system (in F ) for σ(Nt, t ≤ s) and motivate.
