Universität zu Köln
Winter semester 2016 / 2017
Institut für Mathematik
Lecturer: Prof. Dr. A. Drewitz
Assistant: L. Schmitz
To be handed in on November 16th before exercise class.
3rd Homework Probability Theory II
(Martingales, stopping times)
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 3.1
(5 points)
Let σ, τ : (Ω, F, P) → N ∪ {∞} be random variables.
a) Show that τ is a stopping time w.r.t. (Fn ) if and only if
{τ = n} ∈ Fn
for all n ∈ N.
(1 p.)
b) Assume that σ and τ are stopping times w.r.t. (Fn ). Prove that
Fσ∧τ = Fσ ∩ Fτ .
(2 p.)
c) Assume that σ and τ are stopping times w.r.t. (Fn ) with σ ≤ τ .
Show that for every fixed A ∈ Fσ ρ := σ1A + τ 1Ac is a stopping time
w.r.t. (Fn ).
(2 p.)
Exercise 3.2
(4 points)
Let τ : Ω → N be a stopping time w.r.t. (Fn ) such that τ ∈ L . Let (Xn )
be a martingale w.r.t. (Fn ) fulfilling |X1 | ≤ C and |Xn −Xn−1 | ≤ C, n ≥ 2,
for some constant C > 0.
Show Xτ ∈ L1 and E [Xτ ] = E [X1 ].
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Remark: Note that τ need not to be bounded.
Exercise 3.3
(6 points)
Let XP
be a sequence of i.i.d. random variables and denote
1 , X2 , . . . ∈ L
Sn := ni=1 Xi the corresponding random walk. Let A := [M, ∞) for some
constant M > 0 and τA := inf{n ∈ N : Sn ∈ A}. Show that
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a) if E [Xn ] > 0, then P (τA < ∞) = 1;
(2 p.)
b) if E [Xn ] < 0, then P (τA = ∞) > 0.
(4 p.)
Hint: The law of large numbers might be helpful. You don’t need stopping
time- or martingale- techniques here.
Exercise 3.4
(0 points)
Let X1 , X2 , . .P
. ∈ L1 be a sequence of i.i.d. non-negative random variables and
denote Sn := ni=1 Xi the corresponding random walk. Further let τ : Ω → N
be a (finite) stopping time w.r.t. (σ(Sn )). Show that
E [Sτ ] = E [τ ] E [X1 ] .
Hint: Note that τ ∧ k is a bounded stopping time for every k ∈ N.
Remark: With a little bit more effort we could drop the assumption of nonnegativeness of the sequence (Xn ), but would have to assume integrability of
the stopping time τ to prove the same result.
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
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