Universität zu Köln
Winter semester 2016 / 2017
Institut für Mathematik
Lecturer: Prof. Dr. A. Drewitz
Assistant: L. Schmitz
To be handed in on January 18th before exercise class.
10th Homework Probability Theory II
(Modification and indistinguishability, some results about Brownian motion)
If not additionally mentioned, the random variables in every exercise are assumed to be defined on a probability space (Ω, F, P).
Exercise 10.1
(4 points)
Let (Xt ) and (Yt ), t ∈ R, be stochastic processes with state space (R, B(R))
and assume that (Xt ) and (Yt ) are modifications with right-continuous sample
paths, i.e. for all ω ∈ Ω
t 7→ Xt (ω) and t 7→ Yt (ω)
are right-continuous functions.
Show that (Xt ) and (Yt ) are indistinguishable.
Exercise 10.2
(5 points)
Let (Bt ), t ∈ [0, ∞), be a d-dimensional Brownian motion starting in 0 =
(0, . . . , 0)P∈ Rd and let e ∈ Rd with ||e||2 = 1. For a, b ∈ Rd , we denote
a · b := di=1 ai bi the usual inner product on Rd .
Show that (e · Bt ), t ∈ [0, ∞), is a one-dimensional Brownian motion.
Hint for Ex. 10.2 and 10.3: Take a look at characteristic functions.
Exercise 10.3
(6 points)
Let (Bt ), t ∈ [0, ∞), be a one-dimensional Brownian motion and denote Ft =
σ(Bs : s ≤ t). Furthermore i denotes the imaginary unit.
a) Show that (Bt ) and (Bt2 − t) are martingales w.r.t. (Ft ).
(1,5 p.)
is a martingale
b) Let u ∈ R. Show that (Xt ) := exp iuBt + 21 u2 t
w.r.t. (Ft ).
(1,5 p.)
c) Proof the reverse statement of b):
(3 p.)
Let (Yt ), t ∈ [0, ∞), be a real stochastic process with continuous sample paths
and Y0= 0. Furthermore assume that for every u ∈ R
exp iuYt + 21 u2 t is a martingale w.r.t. (F̃t ), where F̃t = σ (Ys : s ≤ t).
Show that (Yt ) is a one-dimensional Brownian motion starting in 0.
Unlike the zero-point exercises from the former sheets, the next exercise can
be handed in, too. It serves as a bonus exercise, i.e. the points gained can be
added to your total points, but they are not included in the total number of
points neccessary to take part on the exam.
Exercise 10.4
(5 extra-points)
Let T ⊂ R and (Xt ), t ∈ T , be a family of real random variables on (Ω, F, P).
Let A ∈ σ(Xt : t ∈ T ).
a) Show that if for ω 0 ∈ Ω and ω ∈ A we have Xt (ω 0 ) = Xt (ω) for all t ∈ T ,
then ω 0 ∈ A.
b) One can further show that there exists a countable subset S ⊂ T , such
that A ∈ σ(Xt : t ∈ S). Use this and a) to prove that
C[0, ∞) := {ω : [0, ∞) → R, ω continuous} ∈
/ B(R)[0,∞) ,
i.e. the set of continuous paths is not measurable in the canonical sense.
If you think there are mistakes or inconsistencies, don’t hesitate to contact me:
[email protected]
Remark: 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