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Past exam of the mathematics course of the University of Cambridge
/
2024
/
ii
/
Paper 1
/
28K
/
c
/
Solution
...
Past exam of the mathematics course of the University of Cambridge
2024
ii
Paper 1
28K
c
OurBigBook.com
Words: 37
On the finite state space,
P
t
=
e
tQ
and
E
x
f
(
X
t
)
=
(
P
t
f
)
(
x
)
.
(195)
Therefore
lim
t
↓
0
t
E
x
f
(
X
t
)
−
f
(
x
)
=
(
lim
t
↓
0
t
P
t
−
I
f
)
(
x
)
=
Q
f
(
x
)
.
(196)
The
matrix
semigroup satisfies
d
t
d
P
t
f
=
P
t
Q
f
. Integrating from
0
to
t
gives the backward
integral
equation
E
x
f
(
X
t
)
=
f
(
x
)
+
∫
0
t
E
x
[
Q
f
(
X
s
)]
d
s
.
(197)
Solved by gpt-5.6-sol high.
Ancestors
(11)
C
28K
Paper 1
Ii
2024
Past exam of the mathematics course of the University of Cambridge
Mathematics course of the University of Cambridge
Course of the University of Cambridge
University of Cambridge
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