Exam Probability II
Prof. Ashkan Nikeghbali
04/02/2013
Exercise 1. A centered continuous Gaussian process (Xt )0≤t≤1 , defined for 0 ≤ t ≤
1, is called a Brownian bridge if its covariance function Γ(s, t) is given by
Γ(s, t) = s(1 − t),
for s ≤ t.
1. What is the variance of Xt for t ∈ [0, 1]?
2. If (Wt )t≥0 is a standard Brownian motion, show that the process
Xt = Wt − tW1 , 0 ≤ t ≤ 1
is a Brownian bridge.
3. Show that if (Wt )t≥0 is a standard Brownian motion, then the process
t , 0 ≤ t ≤ 1
Xt = (1 − t)W 1−t
is a Brownian bridge.
Exercise 2. Let (Ω, F, (Fn ), P) be a filtered probability space. Let (Yn )n≥1 be a
process such that Yn is Fn−1 measurable for every n and let (Xn )n≥0 be an (Fn )martingale. Consider the stochastic process (Zn )n≥0 defined by Z0 = 0 and
Zn =
n
X
Yk (Xk − Xk−1 ), n ≥ 1.
k=1
Prove that if Xn ∈ L2 for every n and Yn ∈ L2 for every n ≥ 1, then (Zn )n≥0 is a
martingale.
Exercise 3. A process (Mn )n≥0 is said to be of independent increments if for every
n the r.v. Mn+1 − Mn is independent of the sigma-algebra Fn = σ(M0 , M1 , · · · , Mn ).
1. Let (Mn ) be a square integrable (Fn )-martingale with independent increments.
We set σ02 = V ar(M0 ) and for k ≥ 1, σk2 = V ar(Mk − Mk−1 ). Show that
V ar(Mn ) =
n
X
k=1
σk2 .
Exam Probability II
04/02/2013
We write
An =
n
X
σk2 .
k=1
Show that Mn2 − An is a martingale.
2. Now let (Mn )n≥0 be a Gaussian process which is also an (Fn )-martingale. Show
that (Mn )n≥0 has independent increments. Show that for every fixed θ ∈ R,
the process
1 2
θ
Zn = exp θMn − θ An
2
is a martingale. Does-it converge a.s.?
3. Now consider a standard Brownian motion (Wt )t≥0 . Set Mn = Wn . What is
(Znθ ) in this case? What is the almost sure limit
Wn
?
n→∞ n
lim
Compute limn→∞ Znθ in this case.
Exercise 4. Let (Ω, F, (Fn ), P) be a filtered probability space. Let (Xn )n≥0 be a
submartingale which satisfies supn≥0 E[|Xn |] < ∞.
1. Show that for fixed n, the sequence (E[Xp+ |Fn ])p≥n is increasing in p.
2. Let us set
Mn = lim E[Xp+ |Fn ].
p→∞
Show that (Mn ) is a positive integrable martingale.
3. Let us set Yn = Mn −Xn . Show that (Yn ) is positive integrable supermartingale.
We come to the conclusion that every submartingale that is bounded in L1 can be
written as the difference of martingale and a supermartingale, both positive and
integrable (this is Krickeberg’s decomposition).
Exercise 5. Let (Yn )n≥1 be a sequence of independent, integrable, identically distributed r.v.’s. We set E[Y1 ] = m, S0 = 0, F0 = {∅, Ω}, Sn = Y1 + · · · + Yn ,
Fn = σ(Y1 , · · · , Yn ). Let ν be an integrable stopping time.
1. We set Xn = Sn − nm. Show that (Xn )n≥0 is a martingale.
2. Show that for every n,
E[Sν∧n ] = mE[ν ∧ n].
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Exam Probability II
04/02/2013
3. Show that Sν is integrable and that
ESν = mE[ν].
Hint: First consider the case Yn ≥ 0.
Exercise 6. Let (Xn )n≥0 be a sequence of independent r.v.’s which are independent
and Gaussian. Prove that
∞
X
P[
Xn2 < ∞] > 0
n=0
is equivalent to
∞
X
E[Xn2 ] < ∞.
n=0
Exercise 7. Let (Ω, F, (Fn ), P) be a filtered probability space. Let (Xn )n≥1 be the
martingale Xn = Y1 + · · · + Yn where (Yn ) are independent and identically distributed
Bernoulli random variables P[Y1 = 1] = P[Y
maxk≤n Xk .
P1 = −1] = 1/2, and set Sn = P
∞
ϕ(n)
<
∞.
Set
Φ(k)
=
Let ϕ : R+ → R+ be a function such that ∞
j=k ϕ(j).
n=0
Prove that ϕ(Sn )(Sn −Xn )+Φ(Sn ) is a martingale which converges almost surely.
What is this limit?
Exercise 8. Let (Xt )t≥0 be a centered Gaussian process which satisfies E[Xt Xs ] = 0
if t 6= s and E[Xt2 ] = 1. Prove that there exists no measurable version for such a
process.
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