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Martingale Theory
Martingale Theory
3 questions
Difficulty 4-9
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Intermediate
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1 intermediate
2 advanced
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intermediate (4/10)
conceptual
A sequence
(
M
n
)
adapted to a filtration
(
F
n
)
is a martingale if
E
[
M
n
+
1
∣
F
n
]
=
M
n
. Which of the following processes is a martingale?
Hide and think first
A.
The running maximum
M
n
=
max
(
X
1
,
…
,
X
n
)
for any i.i.d. sequence
B.
The cumulative sum
S
n
=
X
1
+
…
+
X
n
where
X
i
are i.i.d. with
E
[
X
i
]
=
1
C.
The cumulative sum
S
n
=
X
1
+
…
+
X
n
where
X
i
are i.i.d. with
E
[
X
i
]
=
0
D.
The sample variance
V
n
=
n
1
∑
i
=
1
n
(
X
i
−
X
ˉ
n
)
2
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