STAT4528, part 2 of the unit
FINAL EXAMINATION
JUNE 2020
Question 1. We will consider a probability space ( ,F ,P), where = [0, 1), F =
B([0, 1)) is a Borel σ-algebra of [0, 1] and P is the Lebesgue measure. Let
Akn =
[
k 1
2n
,
k
2n
)
, k = 1, 2, . . . , 2n, n = 0, 1, . . . .
(a) Show that for Fn = σ (Akn; k = 1, . . . , 2n) we have
F = σ
( ∞
n=0
Fn
)
.
(b) Let
ξn(ω) =
2n∑
k=1
k 1
2n
IAkn(ω), n ≥ 0 ,
and let f : [0, 1]→ R be a bounded measurable function. Let
Xn =
f
(
ξn +
1
2n
) f (ξn)
1/2n
.
Show that (Xn,Fn) is a martingale.
(c) Assume that there exists a constant L > 0 such that
|f(x) f(y)| ≤ L|x y|, x, y ∈ [0, 1] .
Show that there exists a random variable X∞, such that Xn → X∞ a.s. and for
n = 0, 1, . . .
Xn = E (X∞|Fn) , a.s.
Show that there exists a measurable set 0 such that P ( 0) = 1 and
Xn(ω) = E (X∞|Fn) (ω), ω ∈ 0, n ≥ 0 .
(d) Show that
f(x) f(0) =
∫ x
0
X∞(ω) dω, x ∈ [0, 1] .
Question 2. Let ( ,F ,P) be a given probability space with the right-continuous filtra-
tion (Ft), where t ∈ [0,∞). and let τ : → [0,∞) be a stopping time.
(a) Let
τn =
[2nτ ] + 1
2n
, n ≥ 1 ,
where for any real number x we denote by [x] the unique integer such that [x] ≤
x < [x] + 1. Show that τn is a stopping time.
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(b) Let (Xt,Ft) be an adapted right-continuous process and assume that
P(τ <∞) = 1. Show that Xτ is a random variable.
(c) Show that for every a ∈ R the process
Mt(a) = exp
(
aWt a
2t
2
)
, t ≥ 0 ,
is an (Ft)-martingale.
(d) Let
f(x, a) = exp
(
ax a
2t
2
)
and gn(x) =
nf
an
(x, 0) .
From part (c) you know that for 0 ≤ s < t, any A ∈ Fs and any a ∈ R
E (Mt(a)IA) = E (Ms(a)IA) .
Assume that for every n ≥ 1, t ≥ s and A ∈ Fs
n
an
[E (Mt(a)IA)] = E
(
n
an
Mt(a)IA
)
and show that for every n the process gn (Wt) is an (Ft)-martingale.
(e) Use (d) to show that the process
Nt = W
4
t 6tW 2t + 3t2 ,
is an (Ft)-martingale.
(f) Let τ = inf {t ≥ 0 : |Wt| = b} for b > 0. Use the Doob stopping theorem to
find Eτ 2. You may use the fact that Eτ = b2
(g) Justify taking derivative under the expectation assumed in (d).
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