Queen’s University - Math 923
Problem Set #1
Winter 2017
Posted: Monday, 23/1/2017
Due: Monday, 30/1/2017
Note: In all that follows, an underlying probability space (Ω, F, P ) is assumed when not specified.
1. Construct a martingale (Xn )n∈N (on a suitable probability space, with respect to a suitable filtration) such that (Xn )n∈N is not
uniformly integrable but supn∈N E(|Xn |) < ∞.
2. Let (Ft )t∈R+ be a filtration; show that the filtration (Ft+ )t∈R+ is right-continuous.
3. Let S, T be (Ft )t∈R+ -stopping times. Show that
(a) T is FT -measurable.
(b) S ≤ T ⇒ FS ⊂ FT
(c) S ∧ T , S ∨ T are (Ft )t∈R+ -stopping times.
(d) FS∧T = FS ∩ FT .
(e) ∀A ∈ FS : A ∩ {S ≤ T } ∈ FT .
4. Let (Sn )n∈N be a sequence of (Ft )t∈R+ -stopping times.
(a) Assume (Sn )n is a monotonically increasing sequence. Show that limn→∞ Sn is an (Ft )t∈R+ -stopping time.
(b) Assume (Sn )n is a monotonically decreasing sequence. Show that limn→∞ Sn is an (Ft+ )t∈R+ -stopping time.
5. Let X = (Xt )t∈T and Y = (Yt )t∈T be (Ft )t∈T -supermartingales (with T ∈ R+ ).
(a) Show that X ∧ Y = (Xt ∧ Yt )t∈T is an (Ft )t∈T -supermartingale.
(b) Show that if X∞ ∈ L1 (Ω, F, P ) closes X to the right (i.e. E(X∞ |Ft ) ≤ Xt a.s. ∀t ∈ T) and Y∞ ∈ L1 (Ω, F, P ) closes Y to
the right, then X∞ ∧ Y∞ closes X ∧ Y to the right.
6. Let Y be a random variable, and let T be an (Ft )t∈R+ -stopping time. Show that Y is FT -measurable iff Y 1{T ≤t} is Ft -measurable
∀t ∈ R+ .
7. Let X = (Xt )t∈R+ be progressively measurable, and let T be an (Ft )t∈R+ -stopping time. Show that the stopped process X T is
progressively measurable.
8. Let A ⊂ R+ ×Ω; we define, for all ω ∈ Ω, Aω = {t ∈ R+ |(t, ω) ∈ A}; the function DA : Ω → R+ ∪{∞} defined by DA (ω) = inf Aω
for all ω ∈ Ω is called the début of A. Show that if the process X = (Xt )t∈R+ is right-continuous and adapted, and if the filtration
(Ft )t∈R+ is right-continuous, then, for any open subset V of R, the début of the set {X ∈ V } is a stopping time. Does the
conclusion hold if the filtration is not assumed right-continuous ? if X is assumed only left-continuous ?
9. On a suitable probability space (Ω, F, P ), equipped with a suitable filtration (Ft )t∈T , construct a process X and a stopping time
T such that XT is not a random variable.