Assume that the counting processes N 1 and N 2 do not jump simultaneously. Prove that the covariation processes   of the corresponding martingales are both identically equal to zero. Let N(t) be an...

Assume that the counting processes N₁
and N₂
do not jump simultaneously. Prove that the covariation processes
  of the corresponding martingales are both identically equal to zero.

Let N(t) be an inhomogeneous Poisson process with intensity λ(t). Then the number of events N(t)−N(s) in the time interval (s,t] is Poisson distributed with parameter
  and the number of events in disjoint time intervals are independent. Let Ft be generated by N(s) for
 and let

a) Prove that
that is, that M(t) is a martingale.

b) Prove that
  that is, that

  is a martingale. Note that this shows that