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Statistical Methods with Applications to Demography and Life Insurance
book

Statistical Methods with Applications to Demography and Life Insurance

by Estate V. Khmaladze
March 2013
Intermediate to advanced content levelIntermediate to advanced
242 pages
5h 47m
English
Chapman and Hall/CRC
Content preview from Statistical Methods with Applications to Demography and Life Insurance
207
hM
l
i(t) is simply a sum of the quadratic variation of the summands:
hM
l
i(t) = hM
Bl
i(t) + hM
Dl
i(t) +
m
k=1
k6=l
hM
Qlk
i(t) +
m
k=1
k6=l
hM
Qkl
i(t),
which is (16.17).
For hM
l
,M
k
i(t) we have:
if assumption (16.16) is satisfied, then for l 6= k
hM
l
,M
k
i(t) = q
l
p
l
(k)
Z
t
0
P
l
(s)ds + q
k
p
k
(l)
Z
t
0
P
k
(s)ds. (16.19)
Indeed, according to (16.18) the product of martingales M
l
(t)M
k
(t)
equals a sum of products of orthogonal martingales and two squares,
M
2
Qlk
(t) and M
2
Qkl
(t). Therefore
hM
l
,M
k
i(t) = hM
Qlk
i(t) + hM
Qkl
i(t),
which is (16.19).
Exercise. Prove the lemma on orthogonality of the martingales
M
Bl
, M
Dl
, M
Qlk
, l, k = 1, ..., m. One will only need to repeat the
argument for orthogonality of M
B
and M
D
from the first part of this
lecture. 4
So, now we agree that populations P
l
,
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Publisher Resources

ISBN: 9781466505742