Universität zu Köln
Winter semester 2016 / 2017
Institut für Mathematik
Lecturer: Prof. Dr. A. Drewitz
Assistant: L. Schmitz
To be handed in on November 9th before exercise class.
2nd Homework Probability Theory II
(Martingales)
To abbreviate notation, in every exercise we assume the random variables to be
defined on a probability space (Ω, F, P). If nothing more is said, (Fn ) defines
a filtration.
Exercise 2.1
(5 points)
Let X1 , X2 , . . . ∈ L2 be a sequence P
of random variables and denote Var(Xn ) :=
σn2 . We further set S0 := 0, Sn := ni=1 Xi , F0 := {∅, Ω} Fn := σ(X1 , . . . , Xn )
for n ≥ 1 and assume that (Sn ) is a martingale w.r.t. (Fn ). Show that
P
a) there exists a sequence (an ) of real numbers such that Sn2 − ni=1 ai is a
martingale w.r.t. (Fn ) if and only if
2
2
E Xn+1
| Fn = an+1 = σn+1
P-a.s.
for all n ≥ 0,
(A)
(3 p.)
b) (A) is fulfilled, if (Xn ) is an independent sequence,
(1 p.)
c) under (A), (Sn2 ) is a submartingale w.r.t. (Fn ).
(1 p.)
Exercise 2.2
(4 points)
Let Q be another finite measure on (Ω, F). For every n ∈ N we define Fn :=
σ(Anj : j ∈ N), where Anj ∈ F and for every n, {An1 , An2 , . . .} is a set
S
of pairwise disjoint sets with P(Anj ) > 0 for all j ∈ N and ˙ j∈N Anj = Ω.
Furthermore, we suppose Fn ⊂ Fn+1 for all n ≥ 1. We set
Xn :=
∞
X
Q(Anj )
j=1
P(Anj )
1Anj
Show that (Xn ) is a martingale w.r.t. (Fn ).
Hint: Take a look at Exercise 1.1.
for all n ≥ 1.
Exercise 2.3
(6 points)
Let (Xn ) be a martingale w.r.t. (Fn ) and denote by
= (σ(X1 , . . . , Xn ))
the canonical filtration generated by the sequence (Xn ). Further, let (Gn ) and
(Hn ) be filtrations with
(FnX )
FnX ⊂ Gn ⊂ Fn ⊂ Hn .
a) Show that (Xn ) is a martingale w.r.t. (Gn ). In particular, (Xn ) is a martingale w.r.t. its canonical filtration.
(2 p.)
b) Is (Xn ) generally a martingale w.r.t. Hn ? If not, construct a counterexample.
(1 p.)
c) Now suppose that X1 , X
P2 , . . . are i.i.d. random variables with X1 ∼
Uni[−1, 1]. We set Sn := ni=1 Xi and Hn := σ(X1 , . . . , Xn , 1{|Xn+1 |≥1/2} ).
Show that (Sn ) is a martingale w.r.t. (Hn ).
(3 p.)
Exercise 2.4
(0 points)
Let (Xn ) be a martingale w.r.t. (Fn ) and Xn ∈ L2 for all n. Show that for all
1 ≤ n1 ≤ n2 ≤ n3 ≤ n4 we have
E [(Xn2 − Xn1 )(Xn4 − Xn3 )] = 0.
Remark: Please just hand in solely exercises with positive rating. If you need
more than one sheet, please clip together the papers and write your group, your
name and the exercise on top of every sheet.
It is possible to hand in solutions in groups of at most two persons.
Total: 15
2