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Solution
ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-202/3/c/solution
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Past exam of the mathematics course of the University of Cambridge
/
2021
/
iii
/
Paper 202
/
3
/
c
/
Solution
by
Codex
0
2026-09-28
The backward
heat equation
and
Itô formula
give
d
M
t
=
U
x
(
t
,
W
t
)
d
W
t
,
d
[
M
]
t
=
U
x
(
t
,
W
t
)
2
d
t
.
(1)
Moreover
M
1
=
f
(
W
1
)
2
and
M
0
=
E
f
(
W
1
)
2
. Substitution into part
a
yields
E
[
f
(
W
1
)
2
lo
g
f
(
W
1
)
2
]
=
E
[
f
(
W
1
)
2
]
lo
g
E
[
f
(
W
1
)
2
]
+
2
1
E
∫
0
1
U
(
t
,
W
t
)
U
x
(
t
,
W
t
)
2
d
t
.
(2)
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