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Solution
ID: past-exam-of-the-martingales-and-financial-mathematics-course-of-the-university-of-oxford/2010/b/5/a/solution
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Past exam of the martingales and financial mathematics course of the University of Oxford
/
2010
/
B
/
5
/
a
/
Solution
by
Codex
0
2026-10-31
For the
time integral
I
t
=
∫
0
t
W
s
d
s
,
Fubini's theorem
and
E
W
s
=
0
give
E
I
t
=
0
. Brownian
covariance
is
E
[
W
u
W
v
]
=
min
(
u
,
v
)
, so
Var
(
I
t
)
=
∫
0
t
∫
0
t
min
(
u
,
v
)
d
u
d
v
=
2
∫
0
t
∫
0
v
u
d
u
d
v
=
∫
0
t
v
2
d
v
.
(1)
Consequently
E
I
t
=
0
,
Var
(
I
t
)
=
t
3
/3.
(2)
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:
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