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
,