202 Population dynamics
is again a martingale. If one were to think about hMi(t) as a sum of
conditional variances, the process hM
1
,M
2
i(t) could be thought of as
the sum of the conditional covariances E[dM
1
(t)dM
2
(t)|F
t
]. Martin-
gales M
1
(t) and M
2
(t) are called orthogonal if hM
1
,M
2
i(t) = 0 for all
t. From the condition (16.8) on their jump points, then, we have shown
that the martingales M
B
and M
D
are orthogonal.
Now let us return to the equality (16.5). Being a definition of M,
this equality is simply an identity. However, let us assume that M is
given and consider (16.5) as an equation with respect to P: the solution
of
dP(t) = cP(t)dt + dM(t)
is given by
P(t) =
Z
t
0
e
c(t−s)
dM(s) + e
ct
P(0). (16.10)
This representation describes properties of the population ...